Three ratios, one mnemonic: In a right-angled triangle, label the sides relative to the angle θ: opposite, adjacent, hypotenuse (the longest, opposite the right angle). Then SOH-CAH-TOA.
Label first: Mark the hypotenuse (opposite the right angle), then opposite and adjacent relative to your angle — the labels change if the angle does.
Pick the ratio with your two sides: Choose the ratio (sin/cos/tan) that links the angle, the side you want, and a side you know. Then rearrange to make the unknown the subject.
IB-style question — find a side
A right triangle has hypotenuse 10 and an angle of 30°. Find the side opposite the 30° angle.
Step by step
- Opposite and hypotenuse → use sin.
- Rearrange.
Final answer
The opposite side is 5.
Unknown on the bottom?: If the unknown is the denominator (e.g. cos θ = 4/x), multiply up and divide: x = 4/cos θ.
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Use the inverse trig button: If you know two sides, form the right ratio, then take the inverse (sin⁻¹, cos⁻¹, tan⁻¹) to get the angle.
IB-style question — find an angle
A right triangle has opposite side 3 and adjacent side 4. Find θ.
Step by step
- Opposite and adjacent → use tan.
- Inverse tangent.
Final answer
θ ≈ 36.9°.
Calculator mode: Make sure your GDC is in degrees (or radians) to match the question — a mode mismatch is a classic lost mark.
Sides only? Use Pythagoras: If a problem gives two sides and wants the third (no angle), use Pythagoras a² + b² = c². Trig is for when an angle is involved.
IB-style question — two-step
A right triangle has legs 5 and 12. Find the hypotenuse, then the angle opposite the side of length 5.
Step by step
- Hypotenuse by Pythagoras.
- Angle: opposite 5, hypotenuse 13.
Final answer
Hypotenuse 13; angle ≈ 22.6°.
Pythagoras = no angle: Three sides, no angle → Pythagoras. Need or have an angle → SOH-CAH-TOA.