GCSE · Pythagoras & right-angled trigonometry

Exact trig values (sin, cos and tan of 0°, 30°, 45°, 60° and 90°)Exam-style questions

Where the exact values of sin, cos and tan come from: two triangles you can sketch in seconds, half a square and half an equilateral triangle, plus a ladder lying flat or standing upright for 0° and 90°. How to use them to find an exact side in surd form, how to get an angle back from an exact ratio, and how to answer a non-calculator "show that" with exact values.

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20 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The diagram shows a right-angled triangle.Diagram NOT accurately drawn

Work out the value of $x$ . (2 marks)
Level 3 · 2 marksTry it
2
Mei says,

" $\sin 60°$ is twice $\sin 30°$ , because $60$ is twice $30$ ."

Is Mei correct?

Show how you decide.

(2 marks)
Level 3 · 2 marksTry it
3
Here is a question and an incorrect solution.

Question: Triangle $ABC$ has a right angle at $C$ . $AB = 10$ cm and angle $BAC = 60°$ . Work out the exact length of $BC$ .

Solution: $\sin 60° = \dfrac{BC}{10}$ , so $BC = 10 \times \dfrac{1}{2} = 5$ cm
(a) Describe the error made.

(1 mark)
(b) Work out the correct exact length of $BC$ .

(2 marks)
Level 3 · 3 marksTry it
4
A tent pole stands vertically on level ground.

A straight guy rope, $3$ m long, joins the top of the pole to a peg in the ground.

The rope makes an angle of $60°$ with the ground.Not drawn accurately
(a) Work out the distance from the peg to the bottom of the pole.

(2 marks)
(b) Work out the exact height of the pole.

Give your answer in its simplest form.

(2 marks)
Level 3 · 4 marksTry it
5
An equilateral triangle has sides of length $10$ cm.Not drawn accurately

Use trigonometry to show that the height of the triangle is $5\sqrt{3}$ cm.

(2 marks)
Level 3 · 2 marksTry it