GCSE Maths: Trigonometry: finding sides (SOH CAH TOA) (Foundation)
Part of the GCSE Maths pythagoras and trigonometry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Trigonometry: finding sides (SOH CAH TOA)
Pythagoras uses two sides to find the third. But what if you know only one side and one angle? That is where trigonometry helps. It links the angles of a right-angled triangle to the ratios of its sides.
Look at the picture. All three triangles have a 30° angle and a right angle, so they are similar (the same shape, different sizes). Work out opposite ÷ hypotenuse for each: 1 ÷ 2, 2 ÷ 4 and 3 ÷ 6. They all equal 0.5. This is no accident. In any right-angled triangle with a 30° angle, the ratio of the side opposite the angle to the hypotenuse is always 0.5. We call this ratio the sine of the angle, so sin 30° = 0.5.
Naming the sides. We name the sides from the point of view of the angle we are using (call it θ, said “theta”).
- The hypotenuse (H) is opposite the right angle. It is the longest side and it never changes.
- The opposite (O) is across the triangle from θ. It does not touch θ.
- The adjacent (A) is next to θ. “Adjacent” means “next to”. It is the side left over that is not the hypotenuse.
There are three trigonometric ratios. Each compares two sides.
sin θ = OH cos θ = AH tan θ = OA
A memory aid: SOH CAH TOA. Sin = Opposite over Hypotenuse, Cos = Adjacent over Hypotenuse, Tan = Opposite over Adjacent. You can also use the triangle trick below: cover the one you want and read what is left.
The method for finding a side.
- Label the sides O, A, H from the angle θ.
- Choose the ratio: look at the side you know and the side you want, and find the two letters in SOH CAH TOA.
- Write the ratio with the numbers: sin θ = O ÷ H becomes, for example, sin 35° = x ÷ 8.2.
- Rearrange to get x alone. If x is on the top, multiply both sides by the number on the bottom. If x is on the bottom, you will multiply by x and then divide.
- Calculate with your calculator in degree mode and round sensibly.
The shortcut for step 4: if the unknown is on the top you multiply, and if the unknown is on the bottom you divide. Use this as a check but know why it works. You can also check that your answer is sensible. The hypotenuse must be the longest side.
Key ideas
- Label O, A, H from the angle θ. H is opposite the right angle.
- SOH: sin θ = O ÷ H. CAH: cos θ = A ÷ H. TOA: tan θ = O ÷ A.
- Choose the ratio from the two sides involved (one known, one wanted).
- Unknown on top: multiply. Unknown on the bottom: divide.
- Calculator in degree mode.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.8.2 cm is the hypotenuse.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 35° = x8.2
- Multiply both sides by 8.2. x is being divided by 8.2. Multiplying both sides by 8.2 gets x on its own.x = 8.2 × sin 35°
- Use the calculator (degree mode). First find sin 35°, then finish the calculation.sin 35° = 0.57357…
x = 8.2 × 0.57357… = 4.70332… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 4.7 cm
Check. The other acute angle is 90° − 35° = 55°. From that angle the opposite and adjacent sides swap, so we use cos: x = 8.2 × cos 55° = 4.70332…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.12 m is the adjacent.
x is the hypotenuse. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the hypotenuse. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 40° = 12x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × cos 40° = 12
- Divide both sides by cos 40°. This leaves x on its own.x = 12cos 40° = 12 ÷ cos 40°
- Use the calculator (degree mode). First find cos 40°, then finish the calculation.cos 40° = 0.76604…
x = 12 ÷ 0.76604… = 15.66488… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 15.7 m
Check. The other acute angle is 90° − 40° = 50°. From that angle the opposite and adjacent sides swap, so we use sin: x = 12 ÷ sin 50° = 15.66488…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.7.5 cm is the adjacent.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 52° = x7.5
- Multiply both sides by 7.5. x is being divided by 7.5. Multiplying both sides by 7.5 gets x on its own.x = 7.5 × tan 52°
- Use the calculator (degree mode). First find tan 52°, then finish the calculation.tan 52° = 1.27994…
x = 7.5 × 1.27994… = 9.59956… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 9.6 cm
Check. The other acute angle is 90° − 52° = 38°. From that angle the opposite and adjacent sides swap, so we use tan: x = 7.5 ÷ tan 38° = 9.59956…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.14.2 mm is the opposite.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the adjacent. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 27° = 14.2x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × tan 27° = 14.2
- Divide both sides by tan 27°. This leaves x on its own.x = 14.2tan 27° = 14.2 ÷ tan 27°
- Use the calculator (degree mode). First find tan 27°, then finish the calculation.tan 27° = 0.50952…
x = 14.2 ÷ 0.50952… = 27.86906… - Round to 3 s.f.. Give the answer to the accuracy asked, with the unit.x = 27.9 mm
Check. The other acute angle is 90° − 27° = 63°. From that angle the opposite and adjacent sides swap, so we use tan: x = 14.2 × tan 63° = 27.86906…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Watch out
Label the sides from the angle you are using, not from the picture. The same side can be opposite for one angle and adjacent for the other. Also check that your calculator is in degree mode. If sin 30° does not give 0.5, it is not.Exercise 20E Labelling sides and choosing the ratio
Non-calculatorKey words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite side is across from θ. The adjacent side is the remaining one, next to θ.
- Find the right angle and the hypotenuse. The right angle is marked with a square. The hypotenuse is the side that does not touch it.Right angle at A, so the hypotenuse is BC.
- Find the opposite side. θ is at B. The opposite side is across the triangle from B, so it does not touch B.The side that does not touch B is AC, so the opposite is AC.
- Find the adjacent side. The adjacent side is next to θ (it touches B) and it is not the hypotenuse.The side left over that touches B is AB, so the adjacent is AB.
Check. Every side has exactly one name. The three names are different. ✓
Watch out. Opposite and adjacent change if you change the angle. The hypotenuse never changes.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite side is across from θ. The adjacent side is the remaining one, next to θ.
- Find the right angle and the hypotenuse. The right angle is marked with a square. The hypotenuse is the side that does not touch it.Right angle at P, so the hypotenuse is QR.
- Find the opposite side. θ is at Q. The opposite side is across the triangle from Q, so it does not touch Q.The side that does not touch Q is PR, so the opposite is PR.
- Find the adjacent side. The adjacent side is next to θ (it touches Q) and it is not the hypotenuse.The side left over that touches Q is PQ, so the adjacent is PQ.
Check. Every side has exactly one name. The three names are different. ✓
Watch out. Opposite and adjacent change if you change the angle. The hypotenuse never changes.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite side is across from θ. The adjacent side is the remaining one, next to θ.
- Find the right angle and the hypotenuse. The right angle is marked with a square. The hypotenuse is the side that does not touch it.Right angle at X, so the hypotenuse is YZ.
- Find the opposite side. θ is at Y. The opposite side is across the triangle from Y, so it does not touch Y.The side that does not touch Y is XZ, so the opposite is XZ.
- Find the adjacent side. The adjacent side is next to θ (it touches Y) and it is not the hypotenuse.The side left over that touches Y is XY, so the adjacent is XY.
Check. Every side has exactly one name. The three names are different. ✓
Watch out. Opposite and adjacent change if you change the angle. The hypotenuse never changes.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. SOH stands for Sin, Opposite, Hypotenuse.
- Read SOH. S is Sin, O is Opposite, H is Hypotenuse.SOH: sin θ = O ÷ H
- Write it as a fraction. The first letter after S is on the top.sin θ = oppositehypotenuse
Check. Cover S in the triangle and you see O over H. ✓
Watch out. O is on top of the fraction for both sin and tan.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. CAH stands for Cos, Adjacent, Hypotenuse.
- Read CAH. C is Cos, A is Adjacent, H is Hypotenuse.CAH: cos θ = A ÷ H
- Write it as a fraction. The A is on the top.cos θ = adjacenthypotenuse
Check. Cover C in the triangle and you see A over H. ✓
Watch out. The hypotenuse is on the bottom of both sin and cos.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. TOA stands for Tan, Opposite, Adjacent.
- Read TOA. T is Tan, O is Opposite, A is Adjacent.TOA: tan θ = O ÷ A
- Write it as a fraction. The O is on the top.tan θ = oppositeadjacent
Check. Cover T in the triangle and you see O over A. ✓
Watch out. Tan does not use the hypotenuse at all.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the hypotenuse and want the opposite.
- Pick the ratio. Write the letters of the two sides: O and H appear together in SOH.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse)
Check. The ratio uses exactly the two sides involved (hypotenuse and opposite) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the adjacent and want the hypotenuse.
- Pick the ratio. Write the letters of the two sides: A and H appear together in CAH.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse)
Check. The ratio uses exactly the two sides involved (adjacent and hypotenuse) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the opposite and want the adjacent.
- Pick the ratio. Write the letters of the two sides: O and A appear together in TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent)
Check. The ratio uses exactly the two sides involved (opposite and adjacent) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the hypotenuse and want the adjacent.
- Pick the ratio. Write the letters of the two sides: A and H appear together in CAH.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse)
Check. The ratio uses exactly the two sides involved (hypotenuse and adjacent) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, choose the ratio, then write it with the numbers. We do not solve it here.
- Label the sides. Opposite: across from the angle. Adjacent: next to it. Hypotenuse: opposite the right angle.8 cm is the hypotenuse. x is the opposite.
- Choose the ratio. The two sides used decide it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Write the equation. Replace θ with 35°, O with the opposite and H with the hypotenuse.sin 35° = x8
Check. The side on the top of the fraction is the one named first in SOH, CAH or TOA. ✓
Watch out. The unknown can be on the top or the bottom of the fraction. Put each side in its own place.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, choose the ratio, then write it with the numbers. We do not solve it here.
- Label the sides. Opposite: across from the angle. Adjacent: next to it. Hypotenuse: opposite the right angle.12 m is the adjacent. x is the hypotenuse.
- Choose the ratio. The two sides used decide it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Write the equation. Replace θ with 40°, A with the adjacent and H with the hypotenuse.cos 40° = 12x
Check. The side on the top of the fraction is the one named first in SOH, CAH or TOA. ✓
Watch out. The unknown can be on the top or the bottom of the fraction. Put each side in its own place.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, choose the ratio, then write it with the numbers. We do not solve it here.
- Label the sides. Opposite: across from the angle. Adjacent: next to it. Hypotenuse: opposite the right angle.6.4 cm is the opposite. x is the adjacent.
- Choose the ratio. The two sides used decide it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Write the equation. Replace θ with 55°, O with the opposite and A with the adjacent.tan 55° = 6.4x
Check. The side on the top of the fraction is the one named first in SOH, CAH or TOA. ✓
Watch out. The unknown can be on the top or the bottom of the fraction. Put each side in its own place.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, choose the ratio, then write it with the numbers. We do not solve it here.
- Label the sides. Opposite: across from the angle. Adjacent: next to it. Hypotenuse: opposite the right angle.9.5 cm is the hypotenuse. x is the adjacent.
- Choose the ratio. The two sides used decide it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Write the equation. Replace θ with 62°, A with the adjacent and H with the hypotenuse.cos 62° = x9.5
Check. The side on the top of the fraction is the one named first in SOH, CAH or TOA. ✓
Watch out. The unknown can be on the top or the bottom of the fraction. Put each side in its own place.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the right-angled triangle with the angle θ and write each ratio as a fraction of the side lengths.
- Label the sides from θ. The side across from θ is 5, the side next to θ is 12 and the side opposite the right angle is 13.O = 5, A = 12, H = 13
- Write sin θ. SOH: opposite over hypotenuse.sin θ = 513
- Write cos θ. CAH: adjacent over hypotenuse.cos θ = 1213
- Write tan θ. TOA: opposite over adjacent.tan θ = 512
Check. The decimals are sin θ = 0.38461…, cos θ = 0.92307…, tan θ = 0.41666…, and sin θ ÷ cos θ = tan θ. ✓
Watch out. Keep the fraction in the same order: opposite first for sin and tan.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. The opposite and adjacent sides depend on which angle we stand at. Stand at each angle in turn.
- Standing at angle A. The side across from A is BC and the side next to A (not the hypotenuse) is AC.For angle A: opposite = BC = 4 cm, adjacent = AC = 3 cm
- Standing at angle B. The side across from B is AC and the side next to B is BC.For angle B: opposite = AC = 3 cm, adjacent = BC = 4 cm
- Compare. The two names swap. The hypotenuse AB stays the same.The side that is opposite for A is adjacent for B, and the other way round.
Check. Angle A + angle B = 90°, so the two acute angles share the same two shorter sides but swap their names. ✓
Watch out. The hypotenuse is the only side whose name does not change.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the hypotenuse and want the opposite.
- Pick the ratio. Write the letters of the two sides: O and H appear together in SOH.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse)
Check. The ratio uses exactly the two sides involved (hypotenuse and opposite) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the opposite and want the adjacent.
- Pick the ratio. Write the letters of the two sides: O and A appear together in TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent)
Check. The ratio uses exactly the two sides involved (opposite and adjacent) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides from the angle, then pick the ratio that has the two sides involved.
- Label the two sides we are using. Opposite: across from the angle. Adjacent: next to the angle. Hypotenuse: opposite the right angle.We have the hypotenuse and want the adjacent.
- Pick the ratio. Write the letters of the two sides: A and H appear together in CAH.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse)
Check. The ratio uses exactly the two sides involved (hypotenuse and adjacent) and nothing else. ✓
Watch out. Choose the ratio from the two sides involved, never from the angle.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Opposite means across from the angle. Check which sides touch the angle x.
- Find the angle x. It is at the end of the 12 cm side and the 13 cm side.x is where the sides 12 cm and 13 cm meet.
- Label from x. The 13 cm side is opposite the right angle: H. The 12 cm side touches x: A. The 5 cm side is across from x: O.O = 5, A = 12, H = 13
- Compare with Dev. Dev wrote tan x = 5 ÷ 13. Tan is opposite over adjacent, so it should use 5 and 12, not the hypotenuse 13.Correct: tan x = 512
Check. Dev’s fraction 5 ÷ 13 = 0.38461… is sin x, not tan x. The correct tan x = 0.41666…. ✓
Watch out. Tan never uses the hypotenuse.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. The hypotenuse is the longest side of a right-angled triangle. Sin and cos put the shorter side on the top and the hypotenuse on the bottom.
- Look at sin θ. sin θ = O ÷ H, and the opposite side O is shorter than the hypotenuse H.O < H
- Divide. A shorter number divided by a longer number is less than 1.O ÷ H < 1, so sin θ < 1
- Same for cos. cos θ = A ÷ H and A is also shorter than H.A ÷ H < 1, so cos θ < 1
Check. Try it: with the 3, 4, 5 triangle, 3÷5 = 0.6 and 4÷5 = 0.8, both less than 1. ✓
Watch out. This is why a calculator gives an error for sin−1(1.3).
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Work out opposite ÷ hypotenuse for each triangle and compare.
- Triangle 1 (the smallest). The opposite is 1 and the hypotenuse is 2.1 ÷ 2 = 0.5
- Triangle 2. The opposite is 2 and the hypotenuse is 4.2 ÷ 4 = 0.5
- Triangle 3 (the biggest). The opposite is 3 and the hypotenuse is 6.3 ÷ 6 = 0.5
- Say what this shows. The ratio is the same because the triangles are similar. It depends only on the angle.sin 30° = 0.5
Check. A calculator confirms sin 30° = 0.5. ✓
Watch out. Enlarging a triangle does not change its angles, so it does not change the ratios.
Exercise 20F Finding sides with sin, cos and tan
Calculator allowedKey words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the calculator in degree mode. Press the trig button, type the angle, close the bracket and press equals.
- Check the calculator is in degree mode. The screen should show D or DEG. If it shows R or G the answer will be wrong.Mode: DEG
- Type sin 40°. Some calculators need the angle after the button, some before.sin 40° = 0.64278…
- Round to 3 decimal places. Look at the fourth decimal place.sin 40° = 0.643
Check. Sin and cos of an acute angle are between 0 and 1: 0.643 fits.
Watch out. If your answers look odd, check DEG mode. sin 30° should be 0.5.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the calculator in degree mode. Press the trig button, type the angle, close the bracket and press equals.
- Check the calculator is in degree mode. The screen should show D or DEG. If it shows R or G the answer will be wrong.Mode: DEG
- Type cos 25°. Some calculators need the angle after the button, some before.cos 25° = 0.90630…
- Round to 3 decimal places. Look at the fourth decimal place.cos 25° = 0.906
Check. Sin and cos of an acute angle are between 0 and 1: 0.906 fits.
Watch out. If your answers look odd, check DEG mode. sin 30° should be 0.5.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the calculator in degree mode. Press the trig button, type the angle, close the bracket and press equals.
- Check the calculator is in degree mode. The screen should show D or DEG. If it shows R or G the answer will be wrong.Mode: DEG
- Type tan 63°. Some calculators need the angle after the button, some before.tan 63° = 1.96261…
- Round to 3 decimal places. Look at the fourth decimal place.tan 63° = 1.963
Check. Tan of an angle less than 45° is less than 1, and tan of an angle bigger than 45° is bigger than 1. ✓
Watch out. If your answers look odd, check DEG mode. sin 30° should be 0.5.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the calculator in degree mode. Press the trig button, type the angle, close the bracket and press equals.
- Check the calculator is in degree mode. The screen should show D or DEG. If it shows R or G the answer will be wrong.Mode: DEG
- Type sin 72°. Some calculators need the angle after the button, some before.sin 72° = 0.95105…
- Round to 3 decimal places. Look at the fourth decimal place.sin 72° = 0.951
Check. Sin and cos of an acute angle are between 0 and 1: 0.951 fits.
Watch out. If your answers look odd, check DEG mode. sin 30° should be 0.5.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the calculator in degree mode. Press the trig button, type the angle, close the bracket and press equals.
- Check the calculator is in degree mode. The screen should show D or DEG. If it shows R or G the answer will be wrong.Mode: DEG
- Type cos 58°. Some calculators need the angle after the button, some before.cos 58° = 0.52991…
- Round to 3 decimal places. Look at the fourth decimal place.cos 58° = 0.530
Check. Sin and cos of an acute angle are between 0 and 1: 0.530 fits.
Watch out. If your answers look odd, check DEG mode. sin 30° should be 0.5.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use the calculator in degree mode. Press the trig button, type the angle, close the bracket and press equals.
- Check the calculator is in degree mode. The screen should show D or DEG. If it shows R or G the answer will be wrong.Mode: DEG
- Type tan 18°. Some calculators need the angle after the button, some before.tan 18° = 0.32491…
- Round to 3 decimal places. Look at the fourth decimal place.tan 18° = 0.325
Check. Tan of an angle less than 45° is less than 1, and tan of an angle bigger than 45° is bigger than 1. ✓
Watch out. If your answers look odd, check DEG mode. sin 30° should be 0.5.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.10 cm is the hypotenuse.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 30° = x10
- Multiply both sides by 10. x is being divided by 10. Multiplying both sides by 10 gets x on its own.x = 10 × sin 30°
- Use the calculator (degree mode). First find sin 30°, then finish the calculation.sin 30° = 0.5
x = 10 × 0.5 = 5 - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 5.0 cm
Check. The other acute angle is 90° − 30° = 60°. From that angle the opposite and adjacent sides swap, so we use cos: x = 10 × cos 60° = 5, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.7.5 cm is the hypotenuse.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 42° = x7.5
- Multiply both sides by 7.5. x is being divided by 7.5. Multiplying both sides by 7.5 gets x on its own.x = 7.5 × sin 42°
- Use the calculator (degree mode). First find sin 42°, then finish the calculation.sin 42° = 0.66913…
x = 7.5 × 0.66913… = 5.01847… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 5.0 cm
Check. The other acute angle is 90° − 42° = 48°. From that angle the opposite and adjacent sides swap, so we use cos: x = 7.5 × cos 48° = 5.01847…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.12 cm is the hypotenuse.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 65° = x12
- Multiply both sides by 12. x is being divided by 12. Multiplying both sides by 12 gets x on its own.x = 12 × sin 65°
- Use the calculator (degree mode). First find sin 65°, then finish the calculation.sin 65° = 0.90630…
x = 12 × 0.90630… = 10.87569… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 10.9 cm
Check. The other acute angle is 90° − 65° = 25°. From that angle the opposite and adjacent sides swap, so we use cos: x = 12 × cos 25° = 10.87569…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.9.6 cm is the hypotenuse.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 23° = x9.6
- Multiply both sides by 9.6. x is being divided by 9.6. Multiplying both sides by 9.6 gets x on its own.x = 9.6 × sin 23°
- Use the calculator (degree mode). First find sin 23°, then finish the calculation.sin 23° = 0.39073…
x = 9.6 × 0.39073… = 3.75101… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 3.8 cm
Check. The other acute angle is 90° − 23° = 67°. From that angle the opposite and adjacent sides swap, so we use cos: x = 9.6 × cos 67° = 3.75101…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.15 cm is the hypotenuse.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 40° = x15
- Multiply both sides by 15. x is being divided by 15. Multiplying both sides by 15 gets x on its own.x = 15 × cos 40°
- Use the calculator (degree mode). First find cos 40°, then finish the calculation.cos 40° = 0.76604…
x = 15 × 0.76604… = 11.49066… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 11.5 cm
Check. The other acute angle is 90° − 40° = 50°. From that angle the opposite and adjacent sides swap, so we use sin: x = 15 × sin 50° = 11.49066…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.8.2 cm is the hypotenuse.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 55° = x8.2
- Multiply both sides by 8.2. x is being divided by 8.2. Multiplying both sides by 8.2 gets x on its own.x = 8.2 × cos 55°
- Use the calculator (degree mode). First find cos 55°, then finish the calculation.cos 55° = 0.57357…
x = 8.2 × 0.57357… = 4.70332… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 4.7 cm
Check. The other acute angle is 90° − 55° = 35°. From that angle the opposite and adjacent sides swap, so we use sin: x = 8.2 × sin 35° = 4.70332…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.11 cm is the hypotenuse.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 28° = x11
- Multiply both sides by 11. x is being divided by 11. Multiplying both sides by 11 gets x on its own.x = 11 × cos 28°
- Use the calculator (degree mode). First find cos 28°, then finish the calculation.cos 28° = 0.88294…
x = 11 × 0.88294… = 9.71242… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 9.7 cm
Check. The other acute angle is 90° − 28° = 62°. From that angle the opposite and adjacent sides swap, so we use sin: x = 11 × sin 62° = 9.71242…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.6.5 cm is the hypotenuse.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 71° = x6.5
- Multiply both sides by 6.5. x is being divided by 6.5. Multiplying both sides by 6.5 gets x on its own.x = 6.5 × cos 71°
- Use the calculator (degree mode). First find cos 71°, then finish the calculation.cos 71° = 0.32556…
x = 6.5 × 0.32556… = 2.11619… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 2.1 cm
Check. The other acute angle is 90° − 71° = 19°. From that angle the opposite and adjacent sides swap, so we use sin: x = 6.5 × sin 19° = 2.11619…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.8 cm is the adjacent.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 50° = x8
- Multiply both sides by 8. x is being divided by 8. Multiplying both sides by 8 gets x on its own.x = 8 × tan 50°
- Use the calculator (degree mode). First find tan 50°, then finish the calculation.tan 50° = 1.19175…
x = 8 × 1.19175… = 9.53402… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 9.5 cm
Check. The other acute angle is 90° − 50° = 40°. From that angle the opposite and adjacent sides swap, so we use tan: x = 8 ÷ tan 40° = 9.53402…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.12.5 cm is the adjacent.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 33° = x12.5
- Multiply both sides by 12.5. x is being divided by 12.5. Multiplying both sides by 12.5 gets x on its own.x = 12.5 × tan 33°
- Use the calculator (degree mode). First find tan 33°, then finish the calculation.tan 33° = 0.64940…
x = 12.5 × 0.64940… = 8.11759… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 8.1 cm
Check. The other acute angle is 90° − 33° = 57°. From that angle the opposite and adjacent sides swap, so we use tan: x = 12.5 ÷ tan 57° = 8.11759…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.5.4 cm is the adjacent.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 68° = x5.4
- Multiply both sides by 5.4. x is being divided by 5.4. Multiplying both sides by 5.4 gets x on its own.x = 5.4 × tan 68°
- Use the calculator (degree mode). First find tan 68°, then finish the calculation.tan 68° = 2.47508…
x = 5.4 × 2.47508… = 13.36546… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 13.4 cm
Check. The other acute angle is 90° − 68° = 22°. From that angle the opposite and adjacent sides swap, so we use tan: x = 5.4 ÷ tan 22° = 13.36546…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.15 cm is the adjacent.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 20° = x15
- Multiply both sides by 15. x is being divided by 15. Multiplying both sides by 15 gets x on its own.x = 15 × tan 20°
- Use the calculator (degree mode). First find tan 20°, then finish the calculation.tan 20° = 0.36397…
x = 15 × 0.36397… = 5.45955… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 5.5 cm
Check. The other acute angle is 90° − 20° = 70°. From that angle the opposite and adjacent sides swap, so we use tan: x = 15 ÷ tan 70° = 5.45955…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.9 cm is the opposite.
x is the hypotenuse. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the hypotenuse. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 35° = 9x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × sin 35° = 9
- Divide both sides by sin 35°. This leaves x on its own.x = 9sin 35° = 9 ÷ sin 35°
- Use the calculator (degree mode). First find sin 35°, then finish the calculation.sin 35° = 0.57357…
x = 9 ÷ 0.57357… = 15.69102… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 15.7 cm
Check. The other acute angle is 90° − 35° = 55°. From that angle the opposite and adjacent sides swap, so we use cos: x = 9 ÷ cos 55° = 15.69102…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.7 cm is the adjacent.
x is the hypotenuse. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the hypotenuse. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 48° = 7x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × cos 48° = 7
- Divide both sides by cos 48°. This leaves x on its own.x = 7cos 48° = 7 ÷ cos 48°
- Use the calculator (degree mode). First find cos 48°, then finish the calculation.cos 48° = 0.66913…
x = 7 ÷ 0.66913… = 10.46133… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 10.5 cm
Check. The other acute angle is 90° − 48° = 42°. From that angle the opposite and adjacent sides swap, so we use sin: x = 7 ÷ sin 42° = 10.46133…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.5.5 cm is the opposite.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the adjacent. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 62° = 5.5x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × tan 62° = 5.5
- Divide both sides by tan 62°. This leaves x on its own.x = 5.5tan 62° = 5.5 ÷ tan 62°
- Use the calculator (degree mode). First find tan 62°, then finish the calculation.tan 62° = 1.88072…
x = 5.5 ÷ 1.88072… = 2.92440… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 2.9 cm
Check. The other acute angle is 90° − 62° = 28°. From that angle the opposite and adjacent sides swap, so we use tan: x = 5.5 × tan 28° = 2.92440…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.14 cm is the adjacent.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 26° = x14
- Multiply both sides by 14. x is being divided by 14. Multiplying both sides by 14 gets x on its own.x = 14 × tan 26°
- Use the calculator (degree mode). First find tan 26°, then finish the calculation.tan 26° = 0.48773…
x = 14 × 0.48773… = 6.82825… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 6.8 cm
Check. The other acute angle is 90° − 26° = 64°. From that angle the opposite and adjacent sides swap, so we use tan: x = 14 ÷ tan 64° = 6.82825…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.19.4 cm is the hypotenuse.
x is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 36° = x19.4
- Multiply both sides by 19.4. x is being divided by 19.4. Multiplying both sides by 19.4 gets x on its own.x = 19.4 × sin 36°
- Use the calculator (degree mode). First find sin 36°, then finish the calculation.sin 36° = 0.58778…
x = 19.4 × 0.58778… = 11.40303… - Round to 3 s.f.. Give the answer to the accuracy asked, with the unit.x = 11.4 cm
Check. The other acute angle is 90° − 36° = 54°. From that angle the opposite and adjacent sides swap, so we use cos: x = 19.4 × cos 54° = 11.40303…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.3.8 m is the opposite.
x is the hypotenuse. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the hypotenuse. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 57° = 3.8x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × sin 57° = 3.8
- Divide both sides by sin 57°. This leaves x on its own.x = 3.8sin 57° = 3.8 ÷ sin 57°
- Use the calculator (degree mode). First find sin 57°, then finish the calculation.sin 57° = 0.83867…
x = 3.8 ÷ 0.83867… = 4.53098… - Round to 2 d.p.. Give the answer to the accuracy asked, with the unit.x = 4.53 m
Check. The other acute angle is 90° − 57° = 33°. From that angle the opposite and adjacent sides swap, so we use cos: x = 3.8 ÷ cos 33° = 4.53098…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw a sketch. The wall is vertical and the ground is horizontal, so the ladder, wall and ground form a right-angled triangle. The ladder is the hypotenuse.Ladder (H) = 6.5 m, angle with the ground = 68°, height up the wall = h (the side opposite the angle).
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.6.5 m is the hypotenuse.
h is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 68° = h6.5
- Multiply both sides by 6.5. h is being divided by 6.5. Multiplying both sides by 6.5 gets h on its own.h = 6.5 × sin 68°
- Use the calculator (degree mode). First find sin 68°, then finish the calculation.sin 68° = 0.92718…
h = 6.5 × 0.92718… = 6.02669… - Round to 2 d.p.. Give the answer to the accuracy asked, with the unit.h = 6.03 m
Check. The other acute angle is 90° − 68° = 22°. From that angle the opposite and adjacent sides swap, so we use cos: h = 6.5 × cos 22° = 6.02669…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw a sketch. The slope of the slide is the hypotenuse of a right-angled triangle. The height of the slide is the side opposite the 38° angle.Slide length (H) = 3.2 m, angle = 38°, height = h (O).
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.3.2 m is the hypotenuse.
h is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 38° = h3.2
- Multiply both sides by 3.2. h is being divided by 3.2. Multiplying both sides by 3.2 gets h on its own.h = 3.2 × sin 38°
- Use the calculator (degree mode). First find sin 38°, then finish the calculation.sin 38° = 0.61566…
h = 3.2 × 0.61566… = 1.97011… - Round to 2 d.p.. Give the answer to the accuracy asked, with the unit.h = 1.97 m
Check. The other acute angle is 90° − 38° = 52°. From that angle the opposite and adjacent sides swap, so we use cos: h = 3.2 × cos 52° = 1.97011…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw a sketch. The kite string is the hypotenuse. The horizontal ground distance is next to the angle at the ground, so it is the adjacent side.String (H) = 45 m, angle = 52° with the ground, horizontal distance = d (A).
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.45 m is the hypotenuse.
d is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 52° = d45
- Multiply both sides by 45. d is being divided by 45. Multiplying both sides by 45 gets d on its own.d = 45 × cos 52°
- Use the calculator (degree mode). First find cos 52°, then finish the calculation.cos 52° = 0.61566…
d = 45 × 0.61566… = 27.70476… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.d = 27.7 m
Check. The other acute angle is 90° − 52° = 38°. From that angle the opposite and adjacent sides swap, so we use sin: d = 45 × sin 38° = 27.70476…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw a sketch. The ramp is the hypotenuse, so the length we want is the hypotenuse. The 0.6 m rise is opposite the 7° angle.Rise (O) = 0.6 m, angle = 7°, ramp length = L (H).
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.0.6 m is the opposite.
L is the hypotenuse. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the hypotenuse. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 7° = 0.6L
- Multiply both sides by L. L is on the bottom of the fraction. Multiplying by L brings it to the top.L × sin 7° = 0.6
- Divide both sides by sin 7°. This leaves L on its own.L = 0.6sin 7° = 0.6 ÷ sin 7°
- Use the calculator (degree mode). First find sin 7°, then finish the calculation.sin 7° = 0.12186…
L = 0.6 ÷ 0.12186… = 4.92330… - Round to 2 d.p.. Give the answer to the accuracy asked, with the unit.L = 4.92 m
Check. The other acute angle is 90° − 7° = 83°. From that angle the opposite and adjacent sides swap, so we use cos: L = 0.6 ÷ cos 83° = 4.92330…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.12 cm is the hypotenuse.
y is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 34° = y12
- Multiply both sides by 12. y is being divided by 12. Multiplying both sides by 12 gets y on its own.y = 12 × sin 34°
- Use the calculator (degree mode). First find sin 34°, then finish the calculation.sin 34° = 0.55919…
y = 12 × 0.55919… = 6.71031… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.y = 6.7 cm
Check. The other acute angle is 90° − 34° = 56°. From that angle the opposite and adjacent sides swap, so we use cos: y = 12 × cos 56° = 6.71031…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.12 cm is the hypotenuse.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 34° = x12
- Multiply both sides by 12. x is being divided by 12. Multiplying both sides by 12 gets x on its own.x = 12 × cos 34°
- Use the calculator (degree mode). First find cos 34°, then finish the calculation.cos 34° = 0.82903…
x = 12 × 0.82903… = 9.94845… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 9.9 cm
Check. The other acute angle is 90° − 34° = 56°. From that angle the opposite and adjacent sides swap, so we use sin: x = 12 × sin 56° = 9.94845…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Anya used cos, but the sides involved are the hypotenuse and the side opposite the angle. Find the correct ratio.
- Label the sides. The 10 cm side is opposite the right angle: it is H. The side x is across from the 40° angle: it is O.We have H and want O.
- Choose the ratio. O and H together make SOH.sin 40° = x10
- Solve. Multiply both sides by 10.x = 10 × sin 40° = 10 × 0.64278… = 6.42787…
- Compare with Anya. Anya’s cos 40° would give the adjacent side, which is a different length.Anya’s answer: 10 × cos 40° = 7.66044…. The correct answer: x = 6.4 cm.
Check. The side opposite 40° must be shorter than the side opposite the bigger angle (50°): 6.4 < 7.7. ✓
Watch out. Label O, A and H first. It takes ten seconds and stops the wrong ratio.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. First find the two shorter sides with trigonometry, then add all three sides.
- Find the side opposite the 36° angle. We have H = 10 and want O, so use sin.O = 10 × sin 36° = 10 × 0.58778… = 5.87785…
- Find the side next to the 36° angle. We have H = 10 and want A, so use cos.A = 10 × cos 36° = 10 × 0.80901… = 8.09016…
- Add the three sides. The perimeter is the distance all the way round. Use the unrounded values and round once at the end.10 + 5.87785… + 8.09016… = 23.96802…
- Round. To 1 decimal place.Perimeter = 24.0 cm
Check. Pythagoras check: 5.87785…2 + 8.09016…2 = 100.00, which is 102 = 100. ✓
Watch out. Do not round the two sides before adding, or the total can be out by 0.1.