GCSE Maths: Trigonometry: finding angles (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 angles
In the last section we knew an angle and wanted a side. Now we know two sides and want the angle. The method is almost the same, but at the end we use the inverse function.
Think about sin 30° = 0.5. Sin takes an angle (30°) and gives a ratio (0.5). The inverse function goes the other way: it takes the ratio and gives the angle. It is written sin−1 and said “inverse sine” (or “arcsin”). So sin−1(0.5) = 30°. On a calculator you press SHIFT (or 2ndF) and then sin, cos or tan. This is not the same as 1 ÷ sin. The little −1 is just a name.
The method for finding an angle.
- Label the two sides you know as O, A or H, measured from the angle you want.
- Choose the ratio (SOH, CAH or TOA) that uses those two sides.
- Write the ratio as a fraction with the numbers: for example tan θ = 5 ÷ 8.
- Use the inverse button on the calculator: θ = tan−1(5 ÷ 8).
- Round sensibly and write the degree sign.
Do not round the fraction before pressing the inverse button, because the rounding error in the ratio makes the angle less accurate. Use the fraction key, or put the division in brackets.
Two checks help. First, an angle in a right-angled triangle is less than 90°. Second, the biggest side is opposite the biggest angle, so if the opposite side is long the angle should be large. We also use this method when the triangle is hidden in another shape, for example an isosceles triangle cut in half, or a rectangle with its diagonal drawn in.
Key ideas
- To find an angle, use the inverse function: sin−1, cos−1 or tan−1 (SHIFT then sin, cos or tan).
- If sin θ = 0.5 then θ = sin−1(0.5) = 30°.
- Choose the ratio from the two sides you know.
- Do not round the ratio before using the inverse.
- Check: the angle must be less than 90° and the biggest side is opposite the biggest 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, pick SOH, CAH or TOA, write the ratio as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.6.7 cm is the opposite.
10 cm is the hypotenuse. - Choose the ratio. We know the opposite and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = hypotenuse).sin x = 6.710 = 0.67
- Use the inverse function sin−1. We know the value of sin x and want the angle. The inverse button (SHIFT or 2ndF, then sin) turns the ratio back into an angle.x = sin−1(6.710) = sin−1(0.67) = 42.06706…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 42.1°
Check. Work forwards: sin 42.1° = 0.670 and the ratio was 0.670. They agree. ✓
Watch out. To find an angle use the inverse button (sin−1), not sin. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.5 cm is the opposite.
8 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 58 = 0.625
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(58) = tan−1(0.625) = 32.00538…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 32.0°
Check. Work forwards: tan 32.0° = 0.625 and the ratio was 0.625. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.4.5 m is the adjacent.
9 m is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 4.59 = 0.5
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(4.59) = cos−1(0.5) = 60°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 60.0°
Check. Work forwards: cos 60.0° = 0.500 and the ratio was 0.500. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw the line of symmetry. In an isosceles triangle the line of symmetry cuts the base in half at 90° and makes two right-angled triangles. We use one of them.Half the base = 12 ÷ 2 = 6 cm. The slanting side is 10 cm (hypotenuse).
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.6 cm is the adjacent.
10 cm is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 610 = 0.6
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(610) = cos−1(0.6) = 53.13010…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 53.1°
Check. Work forwards: cos 53.1° = 0.600 and the ratio was 0.600. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
Watch out
Use the inverse button, not the normal one. If you type sin(0.67) you get 0.0117…, not the angle. Also remember to put the degree sign in the answer. And if your calculator gives an error for sin−1 or cos−1, check that the number you typed is not bigger than 1: the hypotenuse must be on the bottom.Exercise 20G Finding angles with the inverse functions
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. The inverse function turns a ratio back into an angle. Use SHIFT (or 2nd) and the trig button.
- Write what we know. sin x equals the ratio.sin x = 0.6 = 0.6
- Press SHIFT, then sin, to get sin−1. The calculator now finds the angle whose sin is the number we type.x = sin−1(0.6) = 36.86989…°
- Round to 1 decimal place. x = 36.9°
Check. sin 36.9° = 0.600, which matches 0.600. ✓
Watch out. Use sin−1 (SHIFT sin) to find an angle. Plain sin finds a ratio from an 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. The inverse function turns a ratio back into an angle. Use SHIFT (or 2nd) and the trig button.
- Write what we know. cos x equals the ratio.cos x = 0.25 = 0.25
- Press SHIFT, then cos, to get cos−1. The calculator now finds the angle whose cos is the number we type.x = cos−1(0.25) = 75.52248…°
- Round to 1 decimal place. x = 75.5°
Check. cos 75.5° = 0.250, which matches 0.250. ✓
Watch out. Use cos−1 (SHIFT cos) to find an angle. Plain cos finds a ratio from an 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. The inverse function turns a ratio back into an angle. Use SHIFT (or 2nd) and the trig button.
- Write what we know. tan x equals the ratio.tan x = 1.2 = 1.2
- Press SHIFT, then tan, to get tan−1. The calculator now finds the angle whose tan is the number we type.x = tan−1(1.2) = 50.19442…°
- Round to 1 decimal place. x = 50.2°
Check. tan 50.2° = 1.200, which matches 1.200. ✓
Watch out. Use tan−1 (SHIFT tan) to find an angle. Plain tan finds a ratio from an 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. The inverse function turns a ratio back into an angle. Use SHIFT (or 2nd) and the trig button.
- Write what we know. sin x equals the ratio.sin x = 37 = 0.42857…
- Press SHIFT, then sin, to get sin−1. The calculator now finds the angle whose sin is the number we type.x = sin−1(0.42857…) = 25.37693…°
- Round to 1 decimal place. x = 25.4°
Check. sin 25.4° = 0.429, which matches 0.429. ✓
Watch out. Use sin−1 (SHIFT sin) to find an angle. Plain sin finds a ratio from an 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, pick SOH, CAH or TOA, write the ratio as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.4 cm is the opposite.
9 cm is the hypotenuse. - Choose the ratio. We know the opposite and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = hypotenuse).sin x = 49 = 0.44444…
- Use the inverse function sin−1. We know the value of sin x and want the angle. The inverse button (SHIFT or 2ndF, then sin) turns the ratio back into an angle.x = sin−1(49) = sin−1(0.44444…) = 26.38779…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 26.4°
Check. Work forwards: sin 26.4° = 0.44 and the ratio was 0.44. They agree. ✓
Watch out. To find an angle use the inverse button (sin−1), not sin. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.7.5 cm is the opposite.
12 cm is the hypotenuse. - Choose the ratio. We know the opposite and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = hypotenuse).sin x = 7.512 = 0.625
- Use the inverse function sin−1. We know the value of sin x and want the angle. The inverse button (SHIFT or 2ndF, then sin) turns the ratio back into an angle.x = sin−1(7.512) = sin−1(0.625) = 38.68218…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 38.7°
Check. Work forwards: sin 38.7° = 0.625 and the ratio was 0.625. They agree. ✓
Watch out. To find an angle use the inverse button (sin−1), not sin. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.3.2 cm is the opposite.
5 cm is the hypotenuse. - Choose the ratio. We know the opposite and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = hypotenuse).sin x = 3.25 = 0.64
- Use the inverse function sin−1. We know the value of sin x and want the angle. The inverse button (SHIFT or 2ndF, then sin) turns the ratio back into an angle.x = sin−1(3.25) = sin−1(0.64) = 39.79181…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 39.8°
Check. Work forwards: sin 39.8° = 0.640 and the ratio was 0.640. They agree. ✓
Watch out. To find an angle use the inverse button (sin−1), not sin. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.16 cm is the opposite.
21 cm is the hypotenuse. - Choose the ratio. We know the opposite and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = hypotenuse).sin x = 1621 = 0.76190…
- Use the inverse function sin−1. We know the value of sin x and want the angle. The inverse button (SHIFT or 2ndF, then sin) turns the ratio back into an angle.x = sin−1(1621) = sin−1(0.76190…) = 49.63240…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 49.6°
Check. Work forwards: sin 49.6° = 0.762 and the ratio was 0.762. They agree. ✓
Watch out. To find an angle use the inverse button (sin−1), not sin. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.6 cm is the adjacent.
10 cm is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 610 = 0.6
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(610) = cos−1(0.6) = 53.13010…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 53.1°
Check. Work forwards: cos 53.1° = 0.600 and the ratio was 0.600. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.3.8 cm is the adjacent.
8 cm is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 3.88 = 0.475
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(3.88) = cos−1(0.475) = 61.64064…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 61.6°
Check. Work forwards: cos 61.6° = 0.48 and the ratio was 0.48. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.12 cm is the adjacent.
15 cm is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 1215 = 0.8
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(1215) = cos−1(0.8) = 36.86989…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 36.9°
Check. Work forwards: cos 36.9° = 0.800 and the ratio was 0.800. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.5.5 cm is the adjacent.
7.2 cm is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 5.57.2 = 0.76388…
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(5.57.2) = cos−1(0.76388…) = 40.19175…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 40.2°
Check. Work forwards: cos 40.2° = 0.764 and the ratio was 0.764. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.5 cm is the opposite.
8 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 58 = 0.625
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(58) = tan−1(0.625) = 32.00538…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 32.0°
Check. Work forwards: tan 32.0° = 0.625 and the ratio was 0.625. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.9 cm is the opposite.
4 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 94 = 2.25
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(94) = tan−1(2.25) = 66.03751…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 66.0°
Check. Work forwards: tan 66.0° = 2.25 and the ratio was 2.25. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.7.5 cm is the opposite.
7.5 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 7.57.5 = 1
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(7.57.5) = tan−1(1) = 45°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 45.0°
Check. Work forwards: tan 45.0° = 1.000 and the ratio was 1.000. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.2.4 cm is the opposite.
11 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 2.411 = 0.21818…
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(2.411) = tan−1(0.21818…) = 12.30801…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 12.3°
Check. Work forwards: tan 12.3° = 0.218 and the ratio was 0.218. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.5 cm is the opposite.
12 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 512 = 0.41666…
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(512) = tan−1(0.41666…) = 22.61986…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 22.6°
Check. Work forwards: tan 22.6° = 0.42 and the ratio was 0.42. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw a sketch. The ramp is the hypotenuse. The height it rises is opposite the angle x at the bottom of the ramp.Ramp (H) = 5 m, rise (O) = 0.8 m, angle with the ground = x.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.0.8 m is the opposite.
5 m is the hypotenuse. - Choose the ratio. We know the opposite and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = hypotenuse).sin x = 0.85 = 0.16
- Use the inverse function sin−1. We know the value of sin x and want the angle. The inverse button (SHIFT or 2ndF, then sin) turns the ratio back into an angle.x = sin−1(0.85) = sin−1(0.16) = 9.20689…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 9.2°
Check. Work forwards: sin 9.2° = 0.160 and the ratio was 0.160. They agree. ✓
Watch out. To find an angle use the inverse button (sin−1), not sin. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw a sketch. The ladder is the hypotenuse. The distance 1.6 m along the ground is next to the angle, so it is the adjacent side.Ladder (H) = 6 m, foot of ladder to wall (A) = 1.6 m, angle with the ground = x.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.1.6 m is the adjacent.
6 m is the hypotenuse. - Choose the ratio. We know the adjacent and the hypotenuse, so we use the ratio that contains both. Use SOH CAH TOA.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Write the sides as a fraction (top = adjacent, bottom = hypotenuse).cos x = 1.66 = 0.26666…
- Use the inverse function cos−1. We know the value of cos x and want the angle. The inverse button (SHIFT or 2ndF, then cos) turns the ratio back into an angle.x = cos−1(1.66) = cos−1(0.26666…) = 74.53399…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 74.5°
Check. Work forwards: cos 74.5° = 0.267 and the ratio was 0.267. They agree. ✓
Watch out. To find an angle use the inverse button (cos−1), not cos. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw a sketch. The diagonal of a rectangle cuts it into two right-angled triangles. The 5 cm side is opposite the angle x at the corner, and the 12 cm side is adjacent.O = 5 cm, A = 12 cm.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.5 cm is the opposite.
12 cm is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 512 = 0.41666…
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(512) = tan−1(0.41666…) = 22.61986…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 22.6°
Check. Work forwards: tan 22.6° = 0.42 and the ratio was 0.42. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Think of the gradient as a ratio. A road with gradient 1 in 8 rises 1 m for every 8 m along the horizontal. So the opposite side is 1 and the adjacent side is 8.O = 1, A = 8.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.1 m is the opposite.
8 m is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 18 = 0.125
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(18) = tan−1(0.125) = 7.12501…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 7.1°
Check. Work forwards: tan 7.1° = 0.125 and the ratio was 0.125. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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 as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw the right-angled triangle. From the origin O to the point P(8, 5), go 8 across and 5 up. The line OP is the hypotenuse, 5 is opposite the angle x at the origin and 8 is adjacent.O = 5, A = 8.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.5 is the opposite.
8 is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 58 = 0.625
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(58) = tan−1(0.625) = 32.00538…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 32.0°
Check. Work forwards: tan 32.0° = 0.625 and the ratio was 0.625. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
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. Sine is O ÷ H, and the opposite side can never be longer than the hypotenuse.
- Say what the number means. If sin x = 1.3, then O ÷ H = 1.3, so O would be 1.3 times as long as H.O = 1.3 × H
- Compare with the hypotenuse. The hypotenuse is the longest side of a right-angled triangle, so O must be shorter than H.1.3 × H is longer than H, which is impossible.
- Check with the calculator. The calculator shows an error (Math ERROR) for sin−1(1.3).No angle has a sine bigger than 1.
Check. Sin and cos of any angle are between −1 and 1; for angles in a right-angled triangle they are between 0 and 1. ✓
Watch out. An error from the inverse button usually means the ratio is upside-down.
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. Find the angle from the sides, then see whether it equals 30°.
- Label the sides. The side across from the angle is O = 6 and the hypotenuse is H = 12.O = 6, H = 12
- Choose the ratio. O and H make SOH.sin x = 612 = 0.5
- Use the inverse sine. This turns 0.5 back into an angle.x = sin−1(0.5) = 30°
- State the conclusion. The angle is exactly 30°, as required.
Check. A right-angled triangle with hypotenuse 12 and angle 30° has opposite side 12 × sin 30° = 12 × 0.5 = 6. ✓
Watch out. In a “show that” question the answer is given. You must show how it arises.
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 trigonometry for the two acute angles, then use the angle sum of a triangle as a check.
- Find angle A. At A the opposite side is 5 and the adjacent is 12. TOA.tan A = 512, A = tan−1(0.41666…) = 22.61986…°
- Find angle B. At B the opposite side is 12 and the adjacent is 5. TOA.tan B = 125, B = tan−1(2.4) = 67.38013…°
- The right angle. The angle at C is 90°.C = 90°
- Round and add. The three angles of a triangle add to 180°.A = 22.6°, B = 67.4°, C = 90°
22.6 + 67.4 + 90 = 180.0
Check. The angles add up to 180°. Also the longest side (13) is opposite the biggest angle (90°). ✓
Watch out. After finding two angles, check that all three add to 180°.