Label first, then substitute: Before writing any formula, label the three sides of the right triangle relative to the angle you are working with.
Opposite is across from θ.
Adjacent is next to θ.
Hypotenuse is across from the right angle (always the longest side).
| Ratio | Full name | Formula |
|---|---|---|
| sin θ | Sine | Opposite / Hypotenuse |
| cos θ | Cosine | Adjacent / Hypotenuse |
| tan θ | Tangent | Opposite / Adjacent |
Memory trick: SOH-CAH-TOA: Sine = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj
Interactive: click sin θ, cos θ, or tan θ to highlight the relevant sides.
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The method: 1.
Label all sides (Opp, Adj, Hyp) relative to the known angle. 2.
Choose the ratio that uses the known side and the unknown side. 3.
Rearrange and solve.
Worked example — find unknown side
In a right triangle, the angle θ = 32° and the hypotenuse = 15 cm.
Find the side opposite to θ.
Step by step
- We know Hyp = 15 cm and want Opp. Use sin.
- Rearrange.
Final answer
The opposite side ≈ 7.95 cm.
Calculator mode: Always ensure your calculator is in degree mode for IB problem.
Radian mode will give a completely wrong answer.
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Use the inverse trig function: To find an angle, use the inverse trig function: sin⁻¹, cos⁻¹, or tan⁻¹ (also written arcsin, arcco, arctan).
Worked example — find angle θ
In a right triangle, Opp = 5 cm and Hyp = 13 cm.
Find angle θ.
Step by step
- Opp and Hyp → use sin.
- Apply inverse sine.
Final answer
θ ≈ 22.6°.
Angles should be between 0° and 90°: In a right-angled triangle, all angles are acute.
If your inverse trig gives a negative answer or something > 90°, recheck your labelling.
IB-style question — two-step right-triangle problem
A ladder 5 m long leans against a wall.
The foot of the ladder is 2 m from the base of the wall.
Find the angle the ladder makes with the ground.
Step by step
- Draw a right triangle. Adj = 2 m (ground), Hyp = 5 m (ladder).
- Use cosine.
- Apply inverse cosine.
Final answer
The ladder makes an angle of approximately 66.4° with the ground.
Special angles save time: GDC is allowed on BOTH papers in the IB.
But the special angles 30°, 45°, 60° have exact values (sin 30° = ½, cos 60° = ½, tan 45° = 1, etc.) — memorise them and you save calculator time.
For any other angle, use your GDC.