Trigonometry finds sides and angles in right-angled triangles. Label the sides relative to the angle: the hypotenuse is opposite the right angle, the opposite side faces the angle, and the adjacent side is next to it. Then use SOH CAH TOA: sin θ = O/H, cos θ = A/H and tan θ = O/A. To find an angle, use the inverse functions sin⁻¹, cos⁻¹ or tan⁻¹.
SOH CAH TOA
- sin θ = opposite ÷ hypotenuse
- cos θ = adjacent ÷ hypotenuse
- tan θ = opposite ÷ adjacent
- To find an angle: θ = sin⁻¹, cos⁻¹ or tan⁻¹ of the ratio
Exact values
- sin 30° = ½, cos 60° = ½, tan 45° = 1
- sin 60° = cos 30° = √3/2, sin 45° = cos 45° = √2/2
- tan 30° = √3/3, tan 60° = √3
Labelling the sides
In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle. The other two sides are named relative to the angle θ you are using: the opposite side faces θ and the adjacent side runs from θ to the right angle. Label the sides first; it decides which ratio to use.
Finding a side
Choose the ratio that links the side you know and the side you want. If the opposite is unknown and the adjacent is 12 cm with θ = 40°, tan links them: x = 12 tan 40° = 10.07 cm. If the unknown is on the bottom of the fraction, such as a hypotenuse, rearrange: hypotenuse = adjacent ÷ cos θ.
Finding an angle
When you know two sides, choose the ratio for those sides and use the inverse function. With opposite 7 cm and adjacent 10 cm, tan θ = 0.7, so θ = tan⁻¹(0.7) = 35.0°. Make sure your calculator is in degrees.
Angles of elevation and depression
An angle of elevation is measured upwards from the horizontal, such as looking up at the top of a tree. An angle of depression is measured downwards from the horizontal, such as looking down from a cliff. Draw a sketch with the horizontal line, mark the right angle, and the problem becomes a standard right-angled triangle.
Exact values
Non-calculator questions use the exact values for 0°, 30°, 45°, 60° and 90°. For example, a side opposite 30° in a triangle with hypotenuse 18 cm is 18 × sin 30° = 18 × ½ = 9 cm. Learn the table, or remember it from the two special triangles.
Worked examples
Example 1. In a right-angled triangle, the side adjacent to a 40° angle is 12 cm. Work out the opposite side, x, to 2 decimal places.
- O and A are involved, so use tan: tan 40° = x/12.
- x = 12 tan 40° = 10.069…
Answer: 10.07 cm.
Example 2. The opposite side is 7 cm and the adjacent side is 10 cm. Work out the angle θ, to 1 decimal place.
- tan θ = 7/10 = 0.7.
- θ = tan⁻¹(0.7) = 34.99…
Answer: 35.0°.
Example 3. The adjacent side to a 55° angle is 8 cm. Work out the hypotenuse, to 2 decimal places.
- cos 55° = 8/h, so h = 8 ÷ cos 55°.
- h = 13.947…
Answer: 13.95 cm.
Example 4. From a point 25 m from the base of a tree, the angle of elevation of the top is 32°. Work out the height of the tree, to 2 decimal places.
- The height is opposite 32°, and 25 m is adjacent.
- Height = 25 tan 32° = 15.621…
Answer: 15.62 m.
Example 5. Without a calculator, find the side opposite a 30° angle in a right-angled triangle with hypotenuse 18 cm.
- sin 30° = ½.
- 18 × ½.
Answer: 9 cm.
Common mistakes
- Labelling sides from the wrong angle. Opposite and adjacent depend on which angle you are using.
- Using the wrong ratio. Pick the ratio containing the side you know and the side you want.
- Calculator in radians. Check the calculator is in degree mode.
- Not rearranging for the hypotenuse. If h is on the bottom, h = adjacent ÷ cos θ.
- Using trigonometry in a non-right-angled triangle. SOH CAH TOA only works with a right angle; otherwise use the sine or cosine rule.
How trigonometry is examined
SOH CAH TOA questions appear at both tiers of the 1MA1 and 4MA1 specifications, and exact trig values are tested on non-calculator papers. Questions ask you to:
- find a missing side using sin, cos or tan
- find a missing angle using an inverse function
- solve problems with angles of elevation and depression
- use exact trigonometric values without a calculator
Use the trigonometry worksheet, which includes answers, and the GCSE geometry revision engine for more.
Trigonometry often follows Pythagoras’ theorem. For non-right-angled triangles, see the sine and cosine rules.
Frequently asked questions
What does SOH CAH TOA mean?
sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.
How do you find an angle in a right-angled triangle?
Work out the ratio of two known sides, then use the inverse sin, cos or tan.
What is an angle of elevation?
The angle measured upwards from the horizontal to an object above you.