Trigonometry and Exact Values

1 · Recap: SOH CAH TOA

This page builds on the KS3 page Trigonometric ratios; go there first if the words below are new. Here is the short version. In a right-angled triangle, pick one of the two acute angles (less than 90°) and call it θ ("theta"). The sides are named from θ:

• hypotenuse (H): the longest side, opposite the right angle. It never changes.
• opposite (O): the side across the triangle from θ, not touching it.
• adjacent (A): the side next to θ that is not the hypotenuse.

Why does this work? Every right-angled triangle with the same angle θ is the same shape (they are similar), just bigger or smaller. So the ratio O ÷ H is the same for all of them, and your calculator has those ratios stored. Change the size below and watch the ratios stay fixed; change θ and they change.

2 · Choose, substitute, solve

Every right-angled trig question follows the same four steps:

1. Label the sides H, O, A from the angle you know (or want).
2. Choose the ratio that uses the two sides in the question: the one you know and the one you want. Cross out the third side; it plays no part.
3. Substitute into the ratio, e.g. sin 35° = x ÷ 10.
4. Solve: if x is on top, multiply (x = 10 × sin 35°); if x is underneath, x and the sin swap places (x = 10 ÷ sin 35°). For an angle, use the inverse: θ = sin⁻¹(O ÷ H) (SHIFT then sin on most calculators).

Practise below. The triangle turns round each time, so you have to label it properly rather than rely on where the sides usually are. Give lengths to 3 s.f. (3 significant figures) and angles to 1 d.p. If you go wrong, the feedback tries to name the mistake.

3 · Multi-step problems

GCSE questions often hide two right-angled triangles in one diagram, sharing a side. You can't reach the answer in one step, so you work out the shared side first and then use it in the second triangle. The trick is to look for the side that belongs to both triangles.

Below, AB and angle α (alpha) are known in the left triangle, and angle β (beta) is known in the right triangle. We want BC. Step through the solution; the colours always follow the triangle being used in that step.

4 · Angles of elevation and depression

Two new words, both measured from the horizontal (a flat, level line):

• The angle of elevation is the angle you turn your eyes up through, from looking straight ahead to looking at something higher, such as the top of a cliff.
• The angle of depression is the angle you turn your eyes down through, from looking straight ahead to looking at something lower, such as a boat below the cliff.

Drag the boat (or use the slider). The horizontal line at the cliff top and the sea are parallel, so the line of sight makes a Z shape: the angle of depression from the cliff equals the angle of elevation from the boat (alternate angles). That gives a right-angled triangle with the height as the opposite side and the distance along the sea as the adjacent: tan is usually the ratio you need.

Try it: real problems

5 · Exact values from two special triangles

For most angles, sin, cos and tan are endless decimals: sin 35° = 0.573576…. But for 30°, 45° and 60° the answers are simple fractions and square roots (surds), which you can work out exactly from two triangles. GCSE papers expect you to know these and use them on the non-calculator paper. Step through each derivation.

Two of these values are often written another way. 1/√2 and √2/2 are equal: multiply top and bottom of 1/√2 by √2 and you get √2/2. This is called rationalising the denominator (see the Surds page). In the same way 1/√3 = √3/3. Either form earns the marks.

What about 0° and 90°?

Make the hypotenuse exactly 1. Then sin θ = O ÷ 1 = O and cos θ = A ÷ 1 = A, so the opposite side is sin θ and the adjacent side is cos θ. Squash θ down to 0°: the triangle flattens, the opposite side shrinks to 0 and the adjacent side stretches to 1. Push θ up to 90°: the opposite becomes 1 and the adjacent becomes 0.

So sin 0° = 0, cos 0° = 1, tan 0° = 0 ÷ 1 = 0; and sin 90° = 1, cos 90° = 0. But tan 90° = 1 ÷ 0, and you can't divide by zero: tan 90° is undefined. (As θ gets close to 90°, tan θ gets enormous; try 89° on a calculator.)

6 · The exact values table

Here is everything in one table. You need to know it, and there's a pattern that makes it easy to rebuild in an exam:

• sin for 0°, 30°, 45°, 60°, 90° is √0/2, √1/2, √2/2, √3/2, √4/2, which simplify to 0, ½, √2/2, √3/2, 1.
• cos is the same list backwards (because cos θ = sin(90° − θ): the two acute angles of a right-angled triangle swap opposite and adjacent).
• tan = sin ÷ cos, because (O/H) ÷ (A/H) = O/A.

Fill in the table from memory, then check. Wrong cells turn red.

7 · Using exact values without a calculator

On the non-calculator paper, a triangle with a 30°, 45° or 60° angle is a signal: use the exact value. Write the ratio, replace sin, cos or tan with its exact value, and tidy up. Leave surds as surds: 5√3 is exact, 8.66 is not.

Each question below is new. Pick an answer: wrong choices explain the likely mistake.

Worked examples

Quick check

What you've learned

• Label H, O, A from the angle, choose the ratio that uses the two sides in the question (SOH CAH TOA), substitute, and solve: multiply if x is on top, divide if x is underneath, inverse (sin⁻¹, cos⁻¹, tan⁻¹) for an angle.
• In multi-step problems, find the side shared by two triangles first, and keep its full value until the end.
• Angles of elevation (looking up) and depression (looking down) are measured from the horizontal, and are equal by alternate angles.
• The isosceles right triangle (1, 1, √2) gives sin 45° = cos 45° = √2/2 and tan 45° = 1. Half an equilateral triangle (1, √3, 2) gives sin 30° = cos 60° = ½, sin 60° = cos 30° = √3/2, tan 30° = √3/3 and tan 60° = √3.
• sin 0° = 0, cos 0° = 1, tan 0° = 0; sin 90° = 1, cos 90° = 0, tan 90° is undefined.
• Pattern: sin goes √0/2, √1/2, √2/2, √3/2, √4/2; cos is the same backwards; tan = sin ÷ cos.
• Without a calculator, replace sin, cos or tan by its exact value and leave surds in the answer.