A complete guide to SOH-CAH-TOA — labelling triangle sides, finding missing sides and angles, special angle values, and angles of elevation and depression.
Right-angled trigonometry gives you the tools to find any missing side or angle in a right-angled triangle, using only one known side and one known angle (or two known sides). It underpins navigation, construction, surveying and countless real-world measurement problems where climbing up to measure something directly simply isn't practical. This guide builds the three trigonometric ratios from first principles, shows how to use them to find missing sides and angles, and applies them to angles of elevation and depression.
Trigonometry studies the relationship between the angles and side lengths of triangles. In a right-angled triangle, once one angle (other than the right angle) and one side length are known, every other side and angle can be calculated exactly — without needing to measure them directly.
The hypotenuse is always the longest side, directly opposite the right angle — this label never changes. The other two labels depend on which angle you are working from: the opposite side is across from that angle, and the adjacent side runs alongside it (touching both the angle and the right angle).
Three ratios connect the sides and angles of a right-angled triangle, remembered by the mnemonic SOH-CAH-TOA:
| Ratio | Formula | Mnemonic |
|---|---|---|
| Sine | sin θ = opposite ÷ hypotenuse | SOH |
| Cosine | cos θ = adjacent ÷ hypotenuse | CAH |
| Tangent | tan θ = opposite ÷ adjacent | TOA |
To find a missing side, identify which two sides (relative to the known angle) are involved — one known, one unknown — choose the matching ratio, then rearrange to solve.
A ladder makes a 58° angle with the ground and has a length (hypotenuse) of 4.5 m. Find the height it reaches up a wall (the opposite side).
sin 58° = opposite ÷ 4.5, so opposite = 4.5 × sin 58° = 4.5 × 0.848 = 3.82 m (3 s.f.).
A right-angled triangle has an adjacent side of 7 cm and an angle of 40°. Find the hypotenuse.
cos 40° = 7 ÷ hypotenuse, so hypotenuse = 7 ÷ cos 40° = 7 ÷ 0.766 = 9.14 cm (3 s.f.).
To find a missing angle when two sides are known, use the inverse trigonometric functions: sin⁻¹, cos⁻¹, or tan⁻¹ (also written arcsin, arccos, arctan), found on a calculator using the SHIFT or 2ndF key.
A right-angled triangle has an opposite side of 5 cm and an adjacent side of 8 cm. Find the angle θ between the adjacent side and the hypotenuse.
tan θ = 5 ÷ 8 = 0.625, so θ = tan⁻¹(0.625) = 32.0° (3 s.f.).
Some angles have exact trigonometric values that are worth memorising, since they appear frequently and can be used without a calculator.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
The angle of elevation is measured upward from the horizontal to a point above the observer (e.g. looking up at the top of a building). The angle of depression is measured downward from the horizontal to a point below the observer (e.g. looking down from a cliff). These two angles are always equal for a pair of observers looking directly at each other, since they are alternate angles between the horizontal (parallel) lines.
A person stands 20 m from the base of a tower and measures the angle of elevation to the top as 35°. Find the tower's height.
tan 35° = height ÷ 20, so height = 20 × tan 35° = 20 × 0.700 = 14.0 m (3 s.f.).
Trigonometry and Pythagoras' theorem often work together: trigonometry uses one angle and one side, while Pythagoras uses two known sides without needing any angle. In multi-step problems, one triangle might be solved with Pythagoras, and the resulting side then used in a trigonometric ratio in a second triangle.
A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. Find the remaining side, then the smallest angle.
By Pythagoras: remaining side = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.
The smallest angle is opposite the shortest side (5 cm): sin θ = 5 ÷ 13, so θ = sin⁻¹(0.385) = 22.6° (3 s.f.).
Right-angled trigonometry is used constantly outside the classroom: finding the height of a building from a measured distance and angle, calculating the length of a ramp needed for a given rise and angle, working out how far a ship has travelled from its bearing and speed, and determining safe ladder angles in construction and safety regulations.
Common errors include: mixing up the opposite and adjacent sides relative to the wrong angle; forgetting that the hypotenuse label never changes, no matter which angle is used; using degrees mode versus radians mode incorrectly on a calculator; and rounding too early in a multi-step calculation, which compounds small errors into a larger final mistake.
| Situation | Approach |
|---|---|
| Know an angle and the hypotenuse, want the opposite side | opposite = hypotenuse × sin θ |
| Know an angle and the hypotenuse, want the adjacent side | adjacent = hypotenuse × cos θ |
| Know the opposite and adjacent sides, want the angle | θ = tan⁻¹(opposite ÷ adjacent) |
| Know two sides, want the third (no angle needed) | Use Pythagoras' theorem instead |
Every right-angled trigonometry question reduces to correctly labelling the triangle relative to the given or required angle, then choosing the one ratio (or Pythagoras) that connects the known information to the unknown.
This article is original EDUSAMBAM educational writing. It is designed as a broad right-angled trigonometry resource covering the three core ratios and their real-world applications for confident problem-solving. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: memorise SOH-CAH-TOA and the special angle values, practise labelling triangles relative to different angles, then work through mixed missing-side and missing-angle problems before attempting elevation and depression applications.
20 questions covering SOH-CAH-TOA, finding missing sides and angles, special angle values, and angles of elevation and depression. Answer every question, then submit to see your score instantly.