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.
Interactive: tap sin θ, cos θ or tan θ to highlight which two sides (opp / adj / hyp) each ratio uses.
Interactive diagram
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Label first: Mark the hypotenuse (opposite the right angle), then opposite and adjacent relative to your angle — the labels change if the angle does.
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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.
The opposite side x and the hypotenuse 10 sit either side of the 30° angle — that pair is sin. sin 30° = x ÷ 10, so x = 10 sin 30° = 5.
Interactive diagram
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GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
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°.
We know the two legs — opposite 3 and adjacent 4 — and want the angle, so use tan: tan θ = 3 ÷ 4, θ = tan⁻¹(3/4) ≈ 36.9°.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
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 first gives the hypotenuse √(5² + 12²) = 13. Then the angle uses the opposite (5) and that hypotenuse (13): sin θ = 5 ÷ 13 → θ ≈ 22.6°.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Pythagoras = no angle: Three sides, no angle → Pythagoras. Need or have an angle → SOH-CAH-TOA.