Exact Trig Values GCSE Questions and Worked Answers

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A ratio, not a rounded calculator result

What you need to know about exact trig values

Start with a triangle containing a 90° angle and choose one of its other angles, called θ (theta). The side opposite the 90° angle is the hypotenuse. Relative to θ, the side across from it is the opposite side and the remaining side beside it is the adjacent side. Sine, cosine and tangent compare pairs of these sides. An exact value keeps that comparison as a whole number, fraction or square root instead of rounding it to a decimal.

Special triangle → side ratio

Build the values from two triangles

Two special right-angled triangles supply the side lengths. Name the sides relative to θ, then divide the pair required by sine, cosine or tangent.

Choose θthe angle being usedθ is the Greek letter theta

Name the sidesopposite, adjacent, hypotenusethe names depend on θ

Form the ratiosin = opp/hyp · cos = adj/hyp · tan = opp/adjuse the exact side lengths

Cut an equilateral triangle with side length 2 down the middle. Each half has a base of 1 and a hypotenuse of 2, so Pythagoras gives the height as √(4 − 1) = √3. Its angles are 30°, 60° and 90°. For a right-angled triangle with two sides of length 1, the other two angles are both 45° and Pythagoras gives a hypotenuse of √(1 + 1) = √2.

The exact values are ratios from two familiar right-angled triangles, so they can be rebuilt rather than remembered as disconnected facts.

For the endpoint values, imagine a hypotenuse of length 1 turning towards the horizontal. As θ approaches 0°, the opposite side approaches 0 and the adjacent side approaches 1. Towards 90°, those lengths swap. This extends sine and cosine to the endpoints; 0° and 90° are not acute angles in an ordinary right-angled triangle.

Exact trigonometric values for the GCSE special angles
Ratio	0°	30°	45°	60°	90°
sin θ	$0$	$\frac12$	$\frac{\sqrt2}{2}$	$\frac{\sqrt3}{2}$	$1$
cos θ	$1$	$\frac{\sqrt3}{2}$	$\frac{\sqrt2}{2}$	$\frac12$	$0$
tan θ	$0$	$\frac{\sqrt3}{3}$	$1$	$\sqrt3$	undefined

Sine rises across the row while cosine is the same row in reverse. Tangent is sine divided by cosine: for example, tan 30° = 1/√3 = √3/3. Tan 90° is undefined because cos 90° = 0.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

State an exact value

What the problem asks: Write sin, cos or tan of a special angle without a calculator.

How to solve it: Recall the table or rebuild the value from the matching special triangle.

Evaluate an exact expression

What the problem asks: Several trig values are added, multiplied or divided.

How to solve it: Replace each trig value first, then simplify the fractions and surds exactly.

Find an exact side

What the problem asks: A right-angled triangle uses a special angle and asks for an exact length.

How to solve it: Choose the correct SOHCAHTOA ratio, substitute its exact value, then rearrange without converting to a decimal.

Justify a special-triangle result

What the problem asks: Show why an exact length or trig value is true.

How to solve it: Draw the relevant special triangle, label its exact side ratio and form the required trig ratio.

A reliable routine

Method for an exact-value problem

Use this method when the angle is one of the GCSE special angles and the question asks for an exact answer or forbids a calculator. It works because the special triangles have exact side ratios.

Identify the trig ratio and special angle.
Write its exact value before doing any other calculation.
Substitute the fraction or surd into the expression or triangle equation.
Simplify using fraction and surd rules.
Keep the final answer exact and follow the requested form. A terminating decimal such as 0.5 can be exact; a rounded value such as 0.707 cannot replace √2/2.

Check: The exact-value table is assessed across tiers. Calculations that manipulate surds or rationalise denominators are marked Higher only below. Tan 90° has no finite value because tan θ = sin θ ÷ cos θ and cos 90° = 0.

Fully worked

Exact trig values GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Derive sin 30°

2 marks

Question

Use the $1:\sqrt3:2$ triangle to find $\sin30^\circ$ .

For the $30^\circ$ angle, the opposite side is $1$ and the hypotenuse is $2$ .

\sin30^\circ=\frac{\text{opposite}}{\text{hypotenuse}}

\boxed{\sin30^\circ=\frac12}

Example 2

Derive cos 45°

Higher only2 marks

Question

A right-angled isosceles triangle has shorter sides $1$ and hypotenuse $\sqrt2$ . Find $\cos45^\circ$ in rationalised form.

\cos45^\circ=\frac1{\sqrt2}

Multiply the top and bottom by $\sqrt2$ . This multiplies the fraction by $1$ while removing the square root from the denominator.

\frac1{\sqrt2}\times\frac{\sqrt2}{\sqrt2}=\boxed{\frac{\sqrt2}{2}}

Example 3

Evaluate an exact expression

2 marks

Question

Find the exact value of $2\sin30^\circ+\cos60^\circ$ .

2\left(\frac12\right)+\frac12

=1+\frac12

\boxed{\frac32}

Example 4

Find an exact opposite side

3 marks

Question

In a right-angled triangle, the hypotenuse is $14$ cm and an acute angle is $30^\circ$ . Find the side opposite that angle exactly.

Let the opposite side be $x$ .

\sin30^\circ=\frac{x}{14}

\frac12=\frac{x}{14}

x=14\times\frac12

\boxed{x=7\text{ cm}}

Example 5

Find an exact height

Higher only3 marks

Question

A right-angled triangle has adjacent side $5$ cm and angle $60^\circ$ . Find the opposite side exactly.

Let the opposite side be $h$ .

\tan60^\circ=\frac{h}{5}

\sqrt3=\frac{h}{5}

\boxed{h=5\sqrt3\text{ cm}}

10 original questions · total 20 marks

Exact trig values GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 25 minutes · Do not use rounded decimals. Write the exact value first, then simplify fractions and surds fully. · answers start collapsed

1

Recall cosine

1 mark

State the exact value of $\cos60^\circ$ .

Show worked answer

\boxed{\cos60^\circ=\frac12}

2

Recall sine

1 mark

State the exact value of $\sin45^\circ$ .

Show worked answer

\boxed{\sin45^\circ=\frac{\sqrt2}{2}}

3

Recall tangent

1 mark

State the exact value of $\tan30^\circ$ .

Show worked answer

\boxed{\tan30^\circ=\frac{\sqrt3}{3}}

4

Add endpoint values

2 marks

Find the exact value of $\sin90^\circ+\cos0^\circ$ .

Show worked answer

1+1=\boxed{2}

5

Multiply an exact value

2 marks

Find the exact value of $4\cos60^\circ$ .

Show worked answer

4\times\frac12=\boxed{2}

6

Keep a surd exact

Higher only2 marks

Find the exact value of $6\sin45^\circ$ .

Show worked answer

6\times\frac{\sqrt2}{2}

\boxed{3\sqrt2}

7

Multiply two trig values

Higher only2 marks

Find the exact value of $\tan60^\circ\cos60^\circ$ .

Show worked answer

\sqrt3\times\frac12

\boxed{\frac{\sqrt3}{2}}

8

Find a side using sine

3 marks

A right-angled triangle has hypotenuse $18$ cm. Find exactly the side opposite a $30^\circ$ angle.

Show worked answer

\sin30^\circ=\frac{x}{18}

\frac12=\frac{x}{18}

\boxed{x=9\text{ cm}}

9

Find a hypotenuse using cosine

Higher only3 marks

In a right-angled triangle, the side adjacent to a $45^\circ$ angle is $8$ cm. Find the hypotenuse exactly.

Show worked answer

\cos45^\circ=\frac8h

\frac{\sqrt2}{2}=\frac8h

h=\frac{16}{\sqrt2}

\boxed{h=8\sqrt2\text{ cm}}

10

Find an equilateral-triangle height

Higher only3 marks

An equilateral triangle has side length $12$ cm. Find its perpendicular height exactly.

Show worked answer

The height splits the triangle into two right-angled triangles with hypotenuse $12$ and base $6$ .

\sin60^\circ=\frac{h}{12}

\frac{\sqrt3}{2}=\frac{h}{12}

\boxed{h=6\sqrt3\text{ cm}}

Examiner-style feedback

Common exact trig values mistakes

Swapping the 30° and 60° values

For sine, the values grow from 1/2 at 30° to √3/2 at 60°. Cosine runs in the opposite order.

Confusing exactness with a required form

1/√2 and √2/2 are equally exact. If rationalised form is requested, write √2/2. Do not replace either with a rounded decimal.

Using a decimal

0.707… is an approximation. If the question asks for an exact value, keep √2/2.

Writing a value for tan 90°

tan 90° is undefined because it would require division by cos 90° = 0.

30-SECOND RECAP

Choose, carry out, check

Translate the wording before you reach for a rule.

Exact means no rounding.
Use the 1:√3:2 and 1:1:√2 triangles.
Sine increases while cosine reverses.
Substitute the exact ratio before simplifying.

Quick answers

Exact trig values FAQ

Which exact trig values are needed at GCSE?

The standard set uses sine and cosine at 0°, 30°, 45°, 60° and 90°, and tangent at 0°, 30°, 45° and 60°.

Why is tan 90° undefined?

Because tan θ = sin θ ÷ cos θ and cos 90° is zero.

Do exact values need a calculator?

No. They are designed to be recalled or rebuilt from special triangles.

What does exact answer mean?

It means a fraction, integer, π-expression or surd rather than a rounded decimal approximation.

Build connected skills

What to revise next

Use the ratios

Trigonometry

Find unknown sides and angles in right-angled triangles.

Revise trigonometry

Simplify roots

Surds

Calculate with and rationalise exact square-root expressions.

Revise surds

Non-right triangles

Cosine rule

Use cosine to connect three sides and one angle.

Revise cosine rule

Content standards

Curriculum and rights review

Reviewed 5 September 2026 against DfE content G21 and current Pearson Edexcel, AQA and OCR GCSE Mathematics specifications. All questions and diagrams are original.

Official specification references