GCSE Maths: Exact values of sin, cos and tan (Higher)
Part of the GCSE Maths pythagoras and trigonometry chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Exact values of sin, cos and tan
Most trigonometry answers are rounded decimals from a calculator. But for a few special angles, the ratios can be written exactly, using fractions and square roots. You need to know these exact values for 0°, 30°, 45°, 60° and 90°. They come up in non-calculator exam questions.
You do not have to learn them by heart. Two simple triangles give you all of them.
The 45° triangle. Take a square of side 1 and cut it along a diagonal. You get a right-angled triangle with two equal sides of length 1 and two angles of 45°. By Pythagoras the hypotenuse is √(12 + 12) = √2.
The 30° and 60° triangle. Take an equilateral triangle of side 2 and cut it in half down the middle. You get a right-angled triangle with angles 30°, 60° and 90°. The shortest side is 1 (half of the base 2), the hypotenuse is 2, and the third side is √(22 − 12) = √3.
Now use SOH CAH TOA. For example, in the 30°–60°–90° triangle, for the 30° angle the opposite side is 1 and the hypotenuse is 2, so sin 30° = 1÷2. For the 60° angle the opposite side is √3, so sin 60° = √3÷2. For tan 30° we get 1÷√3. We usually remove the square root from the bottom by multiplying the top and bottom by √3, which gives √3÷3. This is called rationalising the denominator.
The values for 0° and 90° come from a triangle that has been squashed flat (0°) or stretched upright (90°). Here is the complete table:
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 12 | √22 | √32 | 1 |
| cos θ | 1 | √32 | √22 | 12 | 0 |
| tan θ | 0 | √33 | 1 | √3 | undefined |
A pattern to help you remember sin: the values are √0÷2, √1÷2, √2÷2, √3÷2, √4÷2, which simplify to 0, ½, √2÷2, √3÷2, 1. The cos row is the sin row backwards. And tan = sin ÷ cos.
Key ideas
- sin 30° = ½ cos 60° = ½ sin 60° = cos 30° = √3÷2 sin 45° = cos 45° = √2÷2.
- tan 30° = √3÷3 (or 1÷√3), tan 45° = 1, tan 60° = √3. tan 90° is undefined.
- sin 0° = 0, cos 0° = 1, tan 0° = 0; sin 90° = 1, cos 90° = 0.
- Rationalise: 1÷√2 = √2÷2 and 1÷√3 = √3÷3.
- Exact answers keep the √ and the fractions. Do not use decimals.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 60° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = √3, adjacent = 1, hypotenuse = 2 (for the 60° angle).
- Write the ratio for sin. SOH: sin = opposite ÷ hypotenuse.sin 60° = √32
- Simplify. Check whether the fraction can be written more simply.sin 60° = √32
Check. Calculator: sin 60° = 0.86602… and the exact value is 0.86602…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 30° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = 1, adjacent = √3, hypotenuse = 2 (for the 30° angle).
- Write the ratio for tan. TOA: tan = opposite ÷ adjacent.tan 30° = 1√3
- Rationalise the denominator. We do not leave a square root on the bottom. Multiply the top and the bottom by √3.1√3 = 1 × √3√3 × √3 = √33
Check. Calculator: tan 30° = 0.57735… and the exact value is 0.57735…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Replace each trigonometric ratio by its exact value, then work out the answer using fractions and surds.
- Write down each exact value. Use the table or the special triangles. Do not use decimals.sin 60° = √32
cos 30° = √32 - Put them into the calculation. Replace each trig value by its exact value.sin 60° × cos 30° = √32 × √32
- Work it out exactly. To multiply fractions, multiply the tops and multiply the bottoms. Then simplify.√32 × √32 = 34
Check. Decimal check: 0.86602… × 0.86602… = 0.75, and 34 = 0.75. ✓
Watch out. Keep surds and fractions. Writing 0.866 for √3 ÷ 2 gives an approximate answer, which does not score full marks in an exact question.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root. 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 SOH CAH TOA as usual, but put in the exact value of sin, cos or tan for the special angle so the answer stays exact.
- Label the sides and choose the ratio. We have the hypotenuse and want the opposite.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers in. The angle is one of the special angles, so use its exact value.sin 60° = x10
sin 60° = √32 - Multiply both sides by 10. x is divided by 10.x = 10 × √32 = 5√3
Check. Decimal check: 10 × sin 60° = 8.66025…, and 5√3 = 8.66025…. ✓
Watch out. If the unknown is on the bottom, dividing by a surd is awkward. Flip the exact value instead and multiply.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Look for the exact value √22 in the cos row of the table of exact values. The angle in that column is the answer.
- Write the equation. We know the exact value of cos x.cos x = √22
- Look along the cos row. Check each angle until the value matches. The values in the row are 1, √32, √22, 12, 0.cos 45° = √22
- Write the angle. An acute angle in a right-angled triangle is between 0° and 90°.x = 45°
Check. Calculator: cos−1(0.70710…) = 45°. ✓
Watch out. This is an exact answer, so give it as a whole number of degrees, not a rounded decimal.
Watch out
Do not confuse the two special values: sin 30° = ½ but sin 60° = √3÷2 (about 0.87). A quick check: sin gets bigger as the angle gets bigger, so sin 60° must be bigger than sin 30°. Also, tan 45° = 1 exactly, and tan 90° has no value.Exercise 20I Exact values of trigonometric ratios
Non-calculatorKey words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 30° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = 1, adjacent = √3, hypotenuse = 2 (for the 30° angle).
- Write the ratio for sin. SOH: sin = opposite ÷ hypotenuse.sin 30° = 12
- Simplify. Check whether the fraction can be written more simply.sin 30° = 12
Check. Calculator: sin 30° = 0.5 and the exact value is 0.5. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 60° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = √3, adjacent = 1, hypotenuse = 2 (for the 60° angle).
- Write the ratio for cos. CAH: cos = adjacent ÷ hypotenuse.cos 60° = 12
- Simplify. Check whether the fraction can be written more simply.cos 60° = 12
Check. Calculator: cos 60° = 0.5 and the exact value is 0.5. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 45° we use the right-angled isosceles triangle with two sides of length 1 (the hypotenuse is √2). These triangles give exact side lengths.Opposite = 1, adjacent = 1, hypotenuse = √2 (for the 45° angle).
- Write the ratio for tan. TOA: tan = opposite ÷ adjacent.tan 45° = 11
- Simplify. Check whether the fraction can be written more simply.tan 45° = 1
Check. Calculator: tan 45° = 1 and the exact value is 1. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. For 0° and 90° picture the right-angled triangle getting thinner and thinner, and read off the ratio.
- Picture the triangle. The angle is 90°, so the triangle is squashed upright. The adjacent side shrinks to length 0 and the opposite side becomes as long as the hypotenuse.Take hypotenuse 1. For 90°: opposite = 1, adjacent = 0.
- Write the ratio for sin. sin = O ÷ Hsin 90° = 11 = 1
- Write the answer. sin 90° = 1
Check. Calculator: sin 90° = 1. ✓
Watch out. tan 90° has no value: a calculator shows an error.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. For 0° and 90° picture the right-angled triangle getting thinner and thinner, and read off the ratio.
- Picture the triangle. The angle is 0°, so the triangle is squashed flat. The opposite side shrinks to length 0 and the adjacent side becomes as long as the hypotenuse.Take hypotenuse 1. For 0°: opposite = 0, adjacent = 1.
- Write the ratio for cos. cos = A ÷ Hcos 0° = 11 = 1
- Write the answer. cos 0° = 1
Check. Calculator: cos 0° = 1. ✓
Watch out. tan 90° has no value: a calculator shows an error.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. For 0° and 90° picture the right-angled triangle getting thinner and thinner, and read off the ratio.
- Picture the triangle. The angle is 0°, so the triangle is squashed flat. The opposite side shrinks to length 0 and the adjacent side becomes as long as the hypotenuse.Take hypotenuse 1. For 0°: opposite = 0, adjacent = 1.
- Write the ratio for tan. tan = O ÷ Atan 0° = 01 = 0
- Write the answer. tan 0° = 0
Check. Calculator: tan 0° = 0. ✓
Watch out. tan 90° has no value: a calculator shows an error.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 60° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = √3, adjacent = 1, hypotenuse = 2 (for the 60° angle).
- Write the ratio for sin. SOH: sin = opposite ÷ hypotenuse.sin 60° = √32
- Simplify. Check whether the fraction can be written more simply.sin 60° = √32
Check. Calculator: sin 60° = 0.86602… and the exact value is 0.86602…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 30° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = 1, adjacent = √3, hypotenuse = 2 (for the 30° angle).
- Write the ratio for cos. CAH: cos = adjacent ÷ hypotenuse.cos 30° = √32
- Simplify. Check whether the fraction can be written more simply.cos 30° = √32
Check. Calculator: cos 30° = 0.86602… and the exact value is 0.86602…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 45° we use the right-angled isosceles triangle with two sides of length 1 (the hypotenuse is √2). These triangles give exact side lengths.Opposite = 1, adjacent = 1, hypotenuse = √2 (for the 45° angle).
- Write the ratio for sin. SOH: sin = opposite ÷ hypotenuse.sin 45° = 1√2
- Rationalise the denominator. We do not leave a square root on the bottom. Multiply the top and the bottom by √2.1√2 = 1 × √2√2 × √2 = √22
Check. Calculator: sin 45° = 0.70710… and the exact value is 0.70710…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 60° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = √3, adjacent = 1, hypotenuse = 2 (for the 60° angle).
- Write the ratio for tan. TOA: tan = opposite ÷ adjacent.tan 60° = √31
- Simplify. Check whether the fraction can be written more simply.tan 60° = √3
Check. Calculator: tan 60° = 1.73205… and the exact value is 1.73205…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 45° we use the right-angled isosceles triangle with two sides of length 1 (the hypotenuse is √2). These triangles give exact side lengths.Opposite = 1, adjacent = 1, hypotenuse = √2 (for the 45° angle).
- Write the ratio for cos. CAH: cos = adjacent ÷ hypotenuse.cos 45° = 1√2
- Rationalise the denominator. We do not leave a square root on the bottom. Multiply the top and the bottom by √2.1√2 = 1 × √2√2 × √2 = √22
Check. Calculator: cos 45° = 0.70710… and the exact value is 0.70710…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Draw the special triangle for the angle, label the sides with exact lengths, then write the ratio.
- Use the special triangle. For 30° we use half of an equilateral triangle of side 2 (sides 1, √3 and 2). These triangles give exact side lengths.Opposite = 1, adjacent = √3, hypotenuse = 2 (for the 30° angle).
- Write the ratio for tan. TOA: tan = opposite ÷ adjacent.tan 30° = 1√3
- Rationalise the denominator. We do not leave a square root on the bottom. Multiply the top and the bottom by √3.1√3 = 1 × √3√3 × √3 = √33
Check. Calculator: tan 30° = 0.57735… and the exact value is 0.57735…. They match. ✓
Watch out. You can learn the table, but if you forget a value you can always rebuild it from the two triangles.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Look for the exact value 12 in the sin row of the table of exact values. The angle in that column is the answer.
- Write the equation. We know the exact value of sin x.sin x = 12
- Look along the sin row. Check each angle until the value matches. The values in the row are 0, 12, √22, √32, 1.sin 30° = 12
- Write the angle. An acute angle in a right-angled triangle is between 0° and 90°.x = 30°
Check. Calculator: sin−1(0.5) = 30°. ✓
Watch out. This is an exact answer, so give it as a whole number of degrees, not a rounded decimal.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Look for the exact value √22 in the cos row of the table of exact values. The angle in that column is the answer.
- Write the equation. We know the exact value of cos x.cos x = √22
- Look along the cos row. Check each angle until the value matches. The values in the row are 1, √32, √22, 12, 0.cos 45° = √22
- Write the angle. An acute angle in a right-angled triangle is between 0° and 90°.x = 45°
Check. Calculator: cos−1(0.70710…) = 45°. ✓
Watch out. This is an exact answer, so give it as a whole number of degrees, not a rounded decimal.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Look for the exact value √3 in the tan row of the table of exact values. The angle in that column is the answer.
- Write the equation. We know the exact value of tan x.tan x = √3
- Look along the tan row. Check each angle until the value matches. The values in the row are 0, √33, 1, √3, undefined.tan 60° = √3
- Write the angle. An acute angle in a right-angled triangle is between 0° and 90°.x = 60°
Check. Calculator: tan−1(1.73205…) = 60°. ✓
Watch out. This is an exact answer, so give it as a whole number of degrees, not a rounded decimal.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Look for the exact value √33 in the tan row of the table of exact values. The angle in that column is the answer.
- Write the equation. We know the exact value of tan x.tan x = √33
- Look along the tan row. Check each angle until the value matches. The values in the row are 0, √33, 1, √3, undefined.tan 30° = √33
- Write the angle. An acute angle in a right-angled triangle is between 0° and 90°.x = 30°
Check. Calculator: tan−1(0.57735…) = 30°. ✓
Watch out. This is an exact answer, so give it as a whole number of degrees, not a rounded decimal.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Replace each trigonometric ratio by its exact value, then work out the answer using fractions and surds.
- Write down each exact value. Use the table or the special triangles. Do not use decimals.sin 30° = 12
cos 60° = 12 - Put them into the calculation. Replace each trig value by its exact value.sin 30° + cos 60° = 12 + 12
- Work it out exactly. To add fractions, give them the same denominator, then add the tops.12 + 12 = 1
Check. Decimal check: 0.5 + 0.5 = 1, and 1 = 1. ✓
Watch out. Keep surds and fractions. Writing 0.866 for √3 ÷ 2 gives an approximate answer, which does not score full marks in an exact question.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Replace each trigonometric ratio by its exact value, then work out the answer using fractions and surds.
- Write down each exact value. Use the table or the special triangles. Do not use decimals.cos 45° = √22
sin 45° = √22 - Put them into the calculation. Replace each trig value by its exact value.cos 45° × sin 45° = √22 × √22
- Work it out exactly. To multiply fractions, multiply the tops and multiply the bottoms. Then simplify.√22 × √22 = 12
Check. Decimal check: 0.70710… × 0.70710… = 0.5, and 12 = 0.5. ✓
Watch out. Keep surds and fractions. Writing 0.866 for √3 ÷ 2 gives an approximate answer, which does not score full marks in an exact question.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Replace each trigonometric ratio by its exact value, then work out the answer using fractions and surds.
- Write down each exact value. Use the table or the special triangles. Do not use decimals.tan 45° = 1
sin 30° = 12 - Put them into the calculation. Replace each trig value by its exact value.tan 45° − sin 30° = 1 − 12
- Work it out exactly. To subtract fractions, give them the same denominator, then subtract the tops.1 − 12 = 12
Check. Decimal check: 1 − 0.5 = 0.5, and 12 = 0.5. ✓
Watch out. Keep surds and fractions. Writing 0.866 for √3 ÷ 2 gives an approximate answer, which does not score full marks in an exact question.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Replace each trigonometric ratio by its exact value, then work out the answer using fractions and surds.
- Write down each exact value. Use the table or the special triangles. Do not use decimals.sin 60° = √32
cos 60° = 12 - Put them into the calculation. Replace each trig value by its exact value.sin 60° ÷ cos 60° = √32 ÷ 12
- Work it out exactly. To divide by a fraction, multiply by its reciprocal (flip it). Then simplify.√32 ÷ 12 = √3
Check. Decimal check: 0.86602… ÷ 0.5 = 1.73205…, and √3 = 1.73205…. ✓
Watch out. Keep surds and fractions. Writing 0.866 for √3 ÷ 2 gives an approximate answer, which does not score full marks in an exact question.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root. 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 SOH CAH TOA as usual, but put in the exact value of sin, cos or tan for the special angle so the answer stays exact.
- Label the sides and choose the ratio. We have the hypotenuse and want the opposite.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers in. The angle is one of the special angles, so use its exact value.sin 30° = x8
sin 30° = 12 - Multiply both sides by 8. x is divided by 8.x = 8 × 12 = 4
Check. Decimal check: 8 × sin 30° = 4, and 4 = 4. ✓
Watch out. If the unknown is on the bottom, dividing by a surd is awkward. Flip the exact value instead and multiply.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root. 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 SOH CAH TOA as usual, but put in the exact value of sin, cos or tan for the special angle so the answer stays exact.
- Label the sides and choose the ratio. We have the hypotenuse and want the adjacent.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers in. The angle is one of the special angles, so use its exact value.cos 30° = y8
cos 30° = √32 - Multiply both sides by 8. y is divided by 8.y = 8 × √32 = 4√3
Check. Decimal check: 8 × cos 30° = 6.92820…, and 4√3 = 6.92820…. ✓
Watch out. If the unknown is on the bottom, dividing by a surd is awkward. Flip the exact value instead and multiply.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root. 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 SOH CAH TOA as usual, but put in the exact value of sin, cos or tan for the special angle so the answer stays exact.
- Label the sides and choose the ratio. We have the hypotenuse and want the opposite.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers in. The angle is one of the special angles, so use its exact value.sin 60° = h6
sin 60° = √32 - Multiply both sides by 6. h is divided by 6.h = 6 × √32 = 3√3
Check. Decimal check: 6 × sin 60° = 5.19615…, and 3√3 = 5.19615…. ✓
Watch out. If the unknown is on the bottom, dividing by a surd is awkward. Flip the exact value instead and multiply.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root. 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 SOH CAH TOA as usual, but put in the exact value of sin, cos or tan for the special angle so the answer stays exact.
- Label the sides and choose the ratio. We have the opposite and want the hypotenuse.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers in. The angle is one of the special angles, so use its exact value.sin 45° = 4x
sin 45° = √22 - Rearrange for x. x is on the bottom, so multiply by x and divide by sin 45°.x = 4 ÷ √22 = 4 × √2
- Simplify. Multiply and rationalise if there is a square root on the bottom.x = 4√2
Check. Decimal check: 4 ÷ sin 45° = 5.65685…, and 4√2 = 5.65685…. ✓
Watch out. If the unknown is on the bottom, dividing by a surd is awkward. Flip the exact value instead and multiply.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Write the triangle’s sides in terms of 1, √3 and 2, then compare with what Eve has done.
- Find sin 30°. The 30° angle has opposite side 1 and hypotenuse 2 in the special triangle.sin 30° = 1 ÷ 2 = ½
- Check Eve’s idea. Eve says sin 60° = 2 × sin 30°. Work out both sides.Left: sin 60° = √3 ÷ 2
Right: 2 × sin 30° = 2 × ½ = 1 - Compare. The two sides are different numbers.√3 ÷ 2 is about 0.866, and this is not 1, so Eve is wrong.
Check. Calculator: sin 60° = 0.86602…, but 2 × sin 30° = 1. ✓
Watch out. Doubling the angle does not double the sine. The graph of sin is a curve, not a straight line.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Work out each side exactly and compare them.
- Left-hand side. Use the exact values of sin 45° and cos 45°.sin 45° + cos 45° = √22 + √22 = √2
- Right-hand side. sin 90° is in the table.sin 90° = 1
- Compare. √2 is about 1.414, which is not 1.The statement is not true.
Check. Calculator: sin 45° + cos 45° = 1.41421…, and sin 90° = 1. ✓
Watch out. Trig functions do not work like multiplication: sin(a + b) is not sin a + sin b.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Use the exact values of sin 60°, sin 30° and cos 30°.
- Left-hand side. Write the exact value of sin 60°.sin 60° = √3 ÷ 2
- Right-hand side. Write the exact values of sin 30° and cos 30° and multiply.2 × sin 30° × cos 30° = 2 × ½ × √3÷2 = 1 × √3÷2 = √3 ÷ 2
- Conclusion. Both sides are equal.So sin 60° = 2 sin 30° cos 30°.
Check. Calculator: sin 60° = 0.86602… and 2 sin 30° cos 30° = 0.86602…. ✓
Watch out. In a “show that” question, work each side out separately and then say they are equal.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Work out the area using exact lengths: area = ½ × (the two shorter sides).
- Find the side opposite the 30° angle. The hypotenuse is 10 and sin 30° = ½.O = 10 × sin 30° = 10 × ½ = 5 cm
- Find the side next to the 30° angle. cos 30° = √3 ÷ 2.A = 10 × cos 30° = 10 × √3÷2 = 5√3 cm
- Use the area formula. Area = ½ × base × height, with the two shorter sides as base and height.Area = ½ × 5 × 5√3 = 25√32 cm2
Check. Decimal check: 25√32 = 21.65063… and ½ × 10 × 10 × sin 30° cos 30° = 21.65063…. ✓
Watch out. Leave your answer exact: do not round √3.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. Use the exact values of sin 60° and cos 60°, square them, and add.
- Write the exact values. From the table.sin 60° = √3 ÷ 2 and cos 60° = ½
- Square each value. Square the top and square the bottom of each fraction.(sin 60°)2 = (√3)222 = 34
(cos 60°)2 = 1222 = 14 - Add. The denominators are the same.34 + 14 = 44 = 1
Check. Calculator: 0.75 + 0.25 = 1. ✓
Watch out. (√3)2 = 3. A square root squared gives back the number.
Key words. An exact value is written with fractions and square roots (surds) instead of a rounded decimal. To rationalise a fraction means to remove the square root from the bottom by multiplying the top and bottom by that square root.
Method. The diagonal makes a 30° angle with the longer side. Use exact trigonometry for both sides, then multiply to get the area.
- Find the length of the rectangle. The diagonal (hypotenuse) is 8 and the length is next to the 30° angle.Length = 8 × cos 30° = 8 × √3÷2 = 4√3 cm
- Find the width. The width is opposite the 30° angle.Width = 8 × sin 30° = 8 × ½ = 4 cm
- Find the area. Area of a rectangle = length × width.Area = 4√3 × 4 = 16√3 cm2
Check. Decimal check: 16√3 = 27.71281…, and 6.92820… × 4 = 27.71281…. ✓
Watch out. Multiply the whole numbers first (4 × 4 = 16), then write the surd.