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Exact trig values

discia.org 10 questions · 26 marks
Name Date
  1. Non-calculator 1 mark

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

    You must not use a calculator.

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    \(\frac{\sqrt{2}}{2}\)
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  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 \(\tan 30^\circ = \frac{\sqrt{3}}{3}\)
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  3. Non-calculator 2 marks

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

    You must not use a calculator.

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    \(2\)
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  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.

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    \(90\)
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  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 \(BC\).

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    (a) \(\frac{1}{2}\)
    (b) \(6\) cm
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  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
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  7. Non-calculator 4 marks

    Show that \(\dfrac{ 26 \cos 60^\circ - \tan 45^\circ }{ 8\cos 30^\circ }\) can be written as \(\tan\, (x + 16)^\circ\), where \((x + 16)^\circ\) is an acute angle.

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

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    \(44\)
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  8. Non-calculator 3 marks

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

    \(\cos x = \frac{\sqrt{3}}{2}\)

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

    \(w = 6x - 4y\)

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

    You must show your working.

    Show answer
    \(4\sqrt{3}\)
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  9. Non-calculator 5 marks
    8√3 cmDCBA60°Diagram NOT accurately drawn

    \(ABCD\) is a trapezium. \(AB\) is parallel to \(DC\).

    Angle \(BCD = 60^\circ\) and \(BC = 8\sqrt{3}\) cm.

    The area of the trapezium is \(180\) cm².

    \(AB : DC = 2 : 3\)

    Find the length of \(DC\).

    You must not use a calculator.

    Show answer
    \(18\ \text{cm}\)
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  10. Non-calculator 1 mark

    Write down the exact value of \(\cos 60^\circ\).

    You must not use a calculator.

    Show answer
    \(\frac{1}{2}\)
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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. \(44\)
  8. \(4\sqrt{3}\)
  9. \(18\ \text{cm}\)
  10. \(\frac{1}{2}\)
discia.org Exact trig values · sheet 1