Trigonometry is one of those topics that scares students before they even try it. But once you understand what it is actually asking you to do it becomes one of the most straightforward topics in mathematics.
This guide covers everything you need to know about trigonometry for your exam — the formulas, the rules and fully worked examples so you can see exactly how to apply them.
What is Trigonometry?
Trigonometry is the study of the relationship between the angles and sides of a triangle. In your exam you will mostly deal with right angled triangles — triangles that have one angle of exactly 90 degrees.
The Three Main Trigonometric Ratios
Everything in basic trigonometry comes down to three ratios. Learn these and you can solve almost any trigonometry question.
| Ratio | Formula | Memory Trick |
|---|---|---|
| Sine | Sin θ = Opposite ÷ Hypotenuse | SOH |
| Cosine | Cos θ = Adjacent ÷ Hypotenuse | CAH |
| Tangent | Tan θ = Opposite ÷ Adjacent | TOA |
Memory trick: SOH CAH TOA Say it out loud ten times and you will never forget it.
Standard Angle Values You Must Memorise
These values come up repeatedly in exams. Memorise them:
| Angle | Sin | Cos | Tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
Worked Example 1: Finding a Missing Side
Question: In a right angled triangle ABC, angle B = 90°, angle A = 35° and the hypotenuse AC = 12cm. Find the length of side BC.
Solution:
Step 1: Identify what you have and what you need.
- Angle A = 35°
- Hypotenuse = 12cm
- BC is opposite angle A
- We need the Opposite side
Step 2: Choose the correct ratio.
- We have Hypotenuse and need Opposite → use SOH
- Sin A = Opposite ÷ Hypotenuse
Step 3: Substitute and solve.
- Sin 35° = BC ÷ 12
- BC = 12 × Sin 35°
- BC = 12 × 0.5736
- BC = 6.88cm
Worked Example 2: Finding a Missing Angle
Question: In a right angled triangle, the opposite side = 8cm and the adjacent side = 6cm. Find the angle θ.
Solution:
Step 1: Identify what you have.
- Opposite = 8cm
- Adjacent = 6cm
- We need the angle
Step 2: Choose the correct ratio.
- We have Opposite and Adjacent → use TOA
- Tan θ = Opposite ÷ Adjacent
Step 3: Substitute and solve.
- Tan θ = 8 ÷ 6
- Tan θ = 1.333
- θ = Tan⁻¹ (1.333)
- θ = 53.1°
Worked Example 3: Angles of Elevation and Depression
This is one of the most common ways trigonometry appears in exams.
Question: A student stands 20 metres from the base of a tree. The angle of elevation to the top of the tree is 40°. Find the height of the tree.
Solution:
Step 1: Draw a diagram.
Step 2: Identify what you have.
- Adjacent = 20m (horizontal distance)
- Opposite = h (height of tree)
- Angle = 40°
Step 3: Choose the correct ratio.
- We have Adjacent and need Opposite → use TOA
- Tan 40° = h ÷ 20
Step 4: Solve.
- h = 20 × Tan 40°
- h = 20 × 0.8391
- h = 16.78 metres
Past Questions
Question 1: From a point P on the ground, the angle of elevation of the top of a building is 25°. If the building is 15 metres tall find the horizontal distance from P to the base of the building. Show all working.
How to answer: Draw a right angled triangle with the building as the vertical side labelled 15m and the horizontal distance as the adjacent side labelled d. The angle of elevation is 25°. Use Tan 25° = 15 ÷ d. Rearrange to get d = 15 ÷ Tan 25°. Substitute Tan 25° = 0.4663. Therefore d = 15 ÷ 0.4663 = 32.18 metres.
Question 2: In triangle XYZ, angle Z = 90°, XZ = 9cm and YZ = 12cm. Calculate: a) The length of XY b) The size of angle X c) The size of angle Y
How to answer:
a) Use Pythagoras theorem first. XY² = XZ² + YZ² XY² = 9² + 12² XY² = 81 + 144 = 225 XY = √225 = 15cm
b) Sin X = Opposite ÷ Hypotenuse = YZ ÷ XY = 12 ÷ 15 = 0.8 Angle X = Sin⁻¹ (0.8) = 53.1°
c) Angle Y = 180° – 90° – 53.1° = 36.9°
Conclusion
Trigonometry is all about three ratios — SOH CAH TOA. Master those three and you can solve any basic trigonometry question your exam throws at you. Practice the worked examples above until the steps feel automatic.
For more mathematics revision visit our [Mathematics subject page] and practise with past questions.
Ready to test yourself? Take our free trigonometry quiz now.
