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.

Start practising Read the lesson

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

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    (a) A right-angled triangle has hypotenuse 88 cm and one angle of 60°60°. Work out the exact length of the side opposite the 60°60° angle. [2]

    (b) Another right-angled triangle has hypotenuse 626\sqrt{2} cm and one angle of 45°45°. Work out the exact length of the side adjacent to the 45°45° angle. [2]
    Level 4 · 4 marksTry it
  2. 2
    The diagram shows a right-angled triangle.9 cmx60°Diagram NOT accurately drawn

    Work out the exact value of xx.

    Give your answer in the form a3a\sqrt{3}, where aa is an integer.

    (3 marks)
    Level 4 · 3 marksTry it
  3. 3
    Show that

    9−6tan⁡30°×sin⁡60°9 - 6\tan 30° \times \sin 60°

    is a whole number.

    (4 marks)
    Level 4 · 4 marksTry it
  4. 4
    Show that

    3tan⁡30°−sin⁡60°2cos⁡45°×sin⁡45°\frac{3\tan 30° - \sin 60°}{2\cos 45° \times \sin 45°}

    can be written as cos⁡y°\cos y°, where 0<y<900 < y < 90

    (4 marks)
    Level 4 · 4 marksTry it
  5. 5
    Work out the exact value of

    6cos⁡30°tan⁡45°+sin⁡30°\frac{6\cos 30°}{\tan 45° + \sin 30°}

    Give your answer in its simplest form.

    (3 marks)
    Level 4 · 3 marksTry it

5 more questions on exact trig values (sin, cos and tan of 0°, 30°, 45°, 60° and 90°)

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free