Learn8300Papers 1–3 · Higher tierGeometry and measuresRight-angled trigonometry

Geometry and measures · Subtopic 11 of 17

Right-Angled Trigonometry

Label the sides opposite, adjacent and hypotenuse relative to the angle. Then SOHCAHTOA tells you which ratio links the two sides you know (or want). Use the ratio to find a side, or the inverse function to find an angle.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Label opposite, adjacent and hypotenuse relative to a given angle.
Choose sine, cosine or tangent and find a missing side.
Use $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ to find a missing angle.
Solve problems with angles of elevation and depression.

Pause and think

Draw several right-angled triangles, all with a $30^\circ$ angle but different sizes. Divide the side opposite $30^\circ$ by the hypotenuse in each. What do you notice?

Section 2 of 14

Core idea

Start hereMeasuring a building from the ground

Stand 30 m from a building and look up to its top at $40^\circ$ . Without climbing, you can find its height: in every right-angled triangle with a $40^\circ$ angle, the sides are in the same ratios. Those ratios are sin, cos and tan.

The big idea

$\sin\theta=\frac{\text{opp}}{\text{hyp}}$ , $\cos\theta=\frac{\text{adj}}{\text{hyp}}$ , $\tan\theta=\frac{\text{opp}}{\text{adj}}$ (SOH CAH TOA). Label from the angle; choose the ratio using the two sides involved.

1. Labelling

Labelling the sides from the angle θ

The hypotenuse is always opposite the right angle. "Opposite" and "adjacent" swap if you use the other angle.

sin θ, cos θ and tan θ from 0° to 90°

sin and cos stay between 0 and 1 because the hypotenuse is the longest side. tan can be any size.

Choosing the ratio

Drawing diagram…

The two sides in the question decide the ratio.

2. Examples

Worked exampleFind sides

(a) hyp 10, angle $35^\circ$ , want opp. (b) opp 7, angle $40^\circ$ , want hyp.

Goal: Two sides.
Strategy: Both use opp and hyp: sin.

$x=10\sin35^\circ=5.74$
Unknown on top: multiply.
$\sin40^\circ=\frac7x\Rightarrow x=\frac7{\sin40^\circ}=10.9$
Unknown on the bottom: divide.

Answer

5.74

;

10.9

✓ CheckHypotenuse

10.9>7

✓.

Worked exampleFind an angle

opp 5, adj 8.

Goal: $\theta$ .
Strategy: tan, then inverse.

$\tan\theta=\tfrac58=0.625$
TOA.
$\theta=\tan^{-1}0.625=32.0^\circ$
Inverse.

Answer

32.0^\circ

✓ Checkopp < adj so

\theta<45^\circ

✓.

Worked exampleAngle of elevation

30 m from a building, elevation $40^\circ$ .

Goal: Height.
Strategy: opp and adj: tan.

$\tan40^\circ=\frac h{30}$
TOA.
$h=30\tan40^\circ=25.2$
Multiply.

Answer

25.2

m (plus eye height if given).

✓ Check

40^\circ<45^\circ

, so height < 30 ✓.

Common misconceptionDegrees mode Check the calculator is in degrees.

Pause and think

Exact value of $\tan45^\circ$ ?

Summary

Label from the angle.
Inverse functions for angles.

Angle of elevation: Angle up from the horizontal.

HistoryThe first trig table. Hipparchus of Nicaea, an astronomer working around

140

BC, made the first known table of what we now call trigonometric values, to predict where the Sun and Moon would be. He is often called the founder of trigonometry.

Fun factSine means "bay". The Sanskrit word jya (bowstring) became the Arabic jiba, which was written without vowels and later read as jaib (a fold or bay). Medieval translators turned that into Latin sinus, which became our "sine". The name is a 1000-year-old translation slip.

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