What you'll learn
This topic covers trigonometry in right-angled triangles using SOH CAH TOA. In this guide you will learn how to label the sides, choose the correct ratio, find a missing side, find a missing angle, and apply trigonometry to real problems such as ladders and angles of elevation. This is a core geometry and measurement skill.
Key terms and definitions
Hypotenuse — the longest side, opposite the right angle.
Opposite — the side opposite the angle you are using.
Adjacent — the side next to the angle (not the hypotenuse).
SOH CAH TOA — sin = O/H, cos = A/H, tan = O/A.
Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) — used to find an angle from a ratio.
Core concepts
Labelling the sides
Relative to the angle you are using, label the hypotenuse (opposite the right angle), the opposite (across from the angle) and the adjacent (next to the angle). Correct labelling decides which ratio to use, so always do it first.
Choosing the ratio
SOH CAH TOA gives the three ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Pick the ratio that uses the two sides you know or want — the one without the unwanted side.
Finding a missing side
Substitute into the ratio and rearrange. If the unknown is on top, multiply; if it is on the bottom, divide. For example, with tan 35° = x/8, x = 8 × tan 35°. Make sure the calculator is in degrees.
Finding a missing angle
When you know two sides, use the inverse function. For sin θ = 4/9, θ = sin⁻¹(4/9). The inverse turns the ratio back into the angle. Again, work in degrees.
Applying to real problems
Trigonometry solves problems with ladders, ramps, heights and distances. Look for the angle of elevation (looking up) or depression (looking down), sketch the right-angled triangle, label the sides and apply SOH CAH TOA.
Worked examples
Example 1: Missing side
Find x where cos 40° = x/10.
x = 10 × cos 40° = 7.66 (3 s.f.).
Example 2: Missing angle
Find θ where tan θ = 5/12.
θ = tan⁻¹(5/12) = 22.6° (3 s.f.).
Example 3: Angle of elevation
A 6 m ladder reaches 5 m up a wall. Find the angle with the ground.
sin θ = 5/6, θ = sin⁻¹(5/6) = 56.4° (3 s.f.).
Common mistakes and how to avoid them
Mislabelling the sides. Opposite and adjacent depend on the chosen angle.
Wrong ratio. Use SOH CAH TOA to match the sides involved.
Calculator in the wrong mode. Set it to degrees.
Forgetting the inverse. Use sin⁻¹/cos⁻¹/tan⁻¹ to find an angle.
Rounding too early. Keep full accuracy until the final answer.
Exam technique for Right-angled Trigonometry
Label hypotenuse, opposite and adjacent for the chosen angle.
Pick the ratio with SOH CAH TOA.
Multiply or divide depending on where the unknown is.
Use the inverse function to find an angle.
Check degrees mode and round only at the end.
Quick revision summary
Right-angled trigonometry uses SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. First label the sides relative to the angle — hypotenuse (opposite the right angle), opposite (across from the angle), adjacent (next to it) — then choose the ratio that uses the sides you know or want. To find a missing side, substitute and rearrange (multiply if the unknown is on top, divide if on the bottom); to find a missing angle, use the inverse functions sin⁻¹, cos⁻¹ or tan⁻¹. Apply it to ladders, ramps and heights, watching for angles of elevation (up) and depression (down). Always keep the calculator in degrees and round only at the end. The common errors are mislabelling the sides, choosing the wrong ratio, wrong calculator mode, forgetting the inverse, and rounding too early. Label, pick the ratio, rearrange, and use inverses for angles.