[discia] GCSE Maths › Geometry & Measures › Exact trig values

Exact trig values

discia.org 10 questions · 27 marks
Name Date
  1. Non-calculator 1 mark

    Write down the exact value of \(\sin 45^\circ\).

    You must not use a calculator.

    Show answer
    \(\frac{\sqrt{2}}{2}\)
    Another like this
  2. Non-calculator 2 marks
    Part of a table of exact trigonometric values is shown below.
    \(x\)\(\sin x\)\(\cos x\)\(\tan x\)
    \(30^\circ\)\(\frac{1}{2}\)
    Complete the row. Give each value exactly. You must not use a calculator.
    Show answer
    \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\tan 30^\circ = \frac{\sqrt{3}}{3}\)
    Another like this
  3. Non-calculator 2 marks

    Work out the exact value of \(\tan 45^\circ + \cos 0^\circ\).

    You must not use a calculator.

    Show answer
    \(2\)
    Another like this
  4. Non-calculator 2 marks

    \(x\) is a number with \(0 \leq x \leq 90\).

    Solve \(\cos x^\circ = 0\).

    You must not use a calculator.

    Show answer
    \(90\)
    Another like this
  5. Non-calculator 3 marks

    \(ABC\) is a right-angled triangle.

    Angle \(ABC = 90^\circ\), angle \(BAC = 30^\circ\) and \(AC = 12\) cm.

    (a)

    State the exact value of \(\sin 30^\circ\).

    (b)

    Work out the exact length of \(BC\).

    Show answer
    (a) \(\frac{1}{2}\)
    (b) \(6\) cm
    Another like this
  6. Non-calculator 3 marks

    Show that the value of \(4\tan 60^\circ \times \cos 30^\circ + 2\sin 30^\circ\) is an integer.

    You must show your working. Do not use a calculator.

    Show answer
    Exact values written, the surds cancel: \(4 \times \frac{3}{2} = 6\) and \(1\) is added, so the value is \(7\), an integer
    Another like this
  7. Non-calculator Higher3 marks

    Show that \(\frac{\cos 60^\circ \sin 60^\circ}{\sin 45^\circ}\) can be written in the form \(\frac{ \sqrt{ a } }{ b }\) where \(a\) and \(b\) are integers.

    You must not use a calculator.

    Show answer
    Working leads to \(\frac{\sqrt{6}}{4}\), so \(a = 6\) and \(b = 4\)
    Another like this
  8. Non-calculator 4 marks

    Show that \(\dfrac{ 10 \sin 60^\circ - \tan 60^\circ }{ 8\cos 60^\circ }\) can be written as \(\tan\, (x + 17)^\circ\), where \((x + 17)^\circ\) is an acute angle.

    You must not use a calculator. State the value of \(x\).

    Show answer
    \(43\)
    Another like this
  9. Non-calculator Higher4 marks
    9 cm4√3 cmDCBA60°

    \(ABCD\) is an isosceles trapezium. \(AB\) is parallel to \(DC\), and \(DC\) is the longer of the two. The diagram gives the length of \(DC\), the length of the slant side \(BC\) and the size of the angle at \(C\).

    Work out the exact area of the trapezium.

    Give your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers.

    You must not use a calculator.

    Show answer
    \(54 - 12\sqrt{3}\) cm²
    Another like this
  10. Non-calculator 3 marks

    \(x\) and \(y\) are acute angles.

    \(\sin x = \frac{1}{2}\)

    \(\tan y = \sqrt{3}\)

    \(w = 4x - y\)

    Work out the exact value of \(4 \sin w\).

    You must show your working.

    Show answer
    \(2\sqrt{3}\)
    Another like this
discia.org Exact trig values · sheet 1
Answers Exact trig values · sheet 1
  1. \(\frac{\sqrt{2}}{2}\)
  2. \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\tan 30^\circ = \frac{\sqrt{3}}{3}\)
  3. \(2\)
  4. \(90\)
  5. (a) \(\frac{1}{2}\)
    (b) \(6\) cm
  6. Exact values written, the surds cancel: \(4 \times \frac{3}{2} = 6\) and \(1\) is added, so the value is \(7\), an integer
  7. Working leads to \(\frac{\sqrt{6}}{4}\), so \(a = 6\) and \(b = 4\)
  8. \(43\)
  9. \(54 - 12\sqrt{3}\) cm²
  10. \(2\sqrt{3}\)
discia.org Exact trig values · sheet 1