GCSE Maths trigonometry
Trigonometry links angles and sides. Here is everything GCSE Maths covers, from SOHCAHTOA to the sine and cosine rules.
On this page
Right-angled triangles: SOHCAHTOA
- sin x = opposite / hypotenuse
- cos x = adjacent / hypotenuse
- tan x = opposite / adjacent
Label the sides relative to the angle, choose the ratio that uses the two sides involved, then solve. Use the inverse functions on your calculator to find an angle.
Exact values
- sin 30° = ½, cos 60° = ½
- sin 60° = cos 30° = √3/2
- sin 45° = cos 45° = √2/2, tan 45° = 1
- tan 30° = 1/√3, tan 60° = √3
- sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0
You need these for non-calculator questions.
Non-right-angled triangles (Higher)
- Sine rule: a/sin A = b/sin B = c/sin C. Use it with a pair of an opposite side and angle.
- Cosine rule: a² = b² + c² − 2bc cos A. Use it with three sides, or two sides and the angle between them.
- Area: ½ab sin C.
Choosing the method
- Right angle? SOHCAHTOA or Pythagoras.
- A side and its opposite angle known? Sine rule.
- Otherwise? Cosine rule.
Common mistakes
- Calculator in radians instead of degrees.
- Labelling opposite and adjacent from the wrong angle.
- Rounding too early in multi-step questions.
See also circle theorems and the full GCSE Maths topic list.
Labelling the triangle first
Before using SOHCAHTOA, label the sides relative to the angle you are working with. The hypotenuse is always the longest side, opposite the right angle. The opposite side is across from your angle. The adjacent side is next to your angle but is not the hypotenuse. Then look at which two sides are involved, one you know and one you want, and choose sin, cos or tan to match. Most mistakes in trigonometry come from skipping this labelling step.
Worked examples
Example 1: finding a side with sine
In a right-angled triangle, the hypotenuse is 10 cm and the angle is 30°. Find the side opposite the angle.
- Opposite and hypotenuse: use sin. sin 30° = O/10.
- O = 10 × sin 30° = 5 cm.
Example 2: finding a side with tangent
The adjacent side is 8 cm and the angle is 40°. Find the opposite side.
- tan 40° = O/8, so O = 8 × tan 40° ≈ 6.71 cm.
Example 3: finding a side on the bottom
The hypotenuse is unknown, the adjacent side is 12 cm and the angle is 35°. Find the hypotenuse.
- cos 35° = 12/H, so H = 12/cos 35° ≈ 14.6 cm. When the unknown is on the bottom, divide.
Example 4: finding an angle
The opposite side is 5 cm and the hypotenuse is 13 cm. Find the angle.
- sin θ = 5/13, so θ = sin−1(5/13) ≈ 22.6°.
Example 5: angle of elevation
From 20 m away, the angle of elevation to the top of a tree is 32°. How tall is the tree?
- tan 32° = h/20, so h = 20 tan 32° ≈ 12.5 m.
Example 6: the sine rule (Higher)
In triangle ABC, angle A = 40°, angle B = 60° and side a = 8 cm. Find side b.
- b/sin B = a/sin A, so b = 8 sin 60°/sin 40° ≈ 10.8 cm.
Example 7: the cosine rule (Higher)
Two sides are 7 cm and 9 cm with an included angle of 50°. Find the third side.
- c² = 49 + 81 − 2 × 7 × 9 × cos 50° ≈ 130 − 81.0 = 49.0.
- c ≈ 7.0 cm.
Example 8: area of a triangle (Higher)
Find the area of a triangle with sides 6 cm and 10 cm and an included angle of 30°.
- Area = ½ab sin C = ½ × 6 × 10 × sin 30° = 15 cm².
Exact values without a calculator
You need to know sin, cos and tan of 0°, 30°, 45°, 60° and 90° for the non-calculator paper. The two triangles in the diagram above let you work them out instead of memorising a table. Half an equilateral triangle with side 2 gives the 30° and 60° values; a right-angled isosceles triangle with shorter sides 1 gives the 45° values. For example, sin 30° = ½, cos 30° = √3/2, tan 45° = 1 and tan 60° = √3.
Practice questions
Try each one before opening the answer.
Q1. Find x if sin x = 0.5 and x is acute.
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Q2. A ladder 6 m long leans against a wall at 70° to the ground. How high up the wall does it reach?
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Q3. Find the angle whose tan is 1.
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Q4. A triangle has sides 5, 7 and 8 cm. Find the angle opposite the 7 cm side.
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Frequently asked questions
How do I know whether to use the sine rule or the cosine rule?
Use the sine rule when you know a side and its opposite angle. Use the cosine rule when you know two sides and the angle between them, or all three sides.
My calculator gives strange answers. Why?
Check it is in degrees mode, usually shown as D or DEG on the screen. Radians mode gives very different answers.
Key takeaways
- Label hypotenuse, opposite and adjacent before you start.
- SOHCAHTOA is for right-angled triangles only.
- Use inverse functions to find angles.
- Learn exact values for 30°, 45° and 60°.
- On Higher, use the sine and cosine rules for other triangles.
Where to go next
Trigonometry links to Pythagoras, surds and vectors.
Trigonometry in 3D (Higher)
Higher papers sometimes ask for a length or angle inside a 3D shape, such as the angle between a diagonal and the base of a cuboid. The method is to find a right-angled triangle inside the shape, draw it separately in 2D, and then use Pythagoras or SOHCAHTOA. Often you need two steps: first find a diagonal across the base with Pythagoras, then use it as one side of a vertical triangle. Drawing the 2D triangle clearly is the key step, and it often earns a method mark by itself.
Trigonometry becomes much easier with regular practice. Do a few questions each week, mixing right-angled and non-right-angled triangles, so that choosing the method becomes automatic.
If you get stuck, sketch the triangle again, mark what you know and what you want, and the right method usually becomes clear.