[discia] GCSE Maths › Geometry & Measures › The Sine Rule

The Sine Rule

discia.org 10 questions · 40 marks
Name Date
  1. Either paper 1 mark

    Here is a triangle.

    x cm16 cm62°51°Diagram NOT accurately drawn

    Circle the correct equation.

    \(\frac{ x }{ \sin 51^\circ } = \frac{ 16 }{ \sin 62^\circ }\)     \(\frac{ x }{ 62^\circ } = \frac{ 16 }{ \sin 51^\circ }\)

    \(\frac{ x }{ \sin 62^\circ } = \frac{ 16 }{ \sin 51^\circ }\)     \(\frac{ \sin 51^\circ }{ x } = \frac{ 16 }{ \sin 62^\circ }\)

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    \(\frac{ x }{ \sin 62^\circ } = \frac{ 16 }{ \sin 51^\circ }\)
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  2. Calculator 3 marks

    \(ABC\) is a triangle.

    7 cmx cmABC65°78°Diagram NOT accurately drawn

    Calculate the length of \(AC\).

    Give your answer to 3 significant figures.

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    \(7.55\ \text{cm}\)
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  3. Calculator 3 marks

    \(ABC\) is a triangle.

    13 cm12 cmABC61°xDiagram NOT accurately drawn

    Calculate the size of the angle marked \(x\).

    Give your answer to 1 decimal place.

    Show answer
    \(53.8^\circ\)
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  4. Calculator 4 marks

    \(ABC\) is a triangle.

    8 cmx cmABC39°44°Diagram NOT accurately drawn
    (a)

    Find the size of angle \(ACB\).

    (b)

    Find the length of \(BC\).

    Give your answer correct to 1 decimal place.

    Show answer
    (a) \(97^\circ\)
    (b) \(5.1\ \text{cm}\)
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  5. Calculator 4 marks

    \(P\) and \(Q\) are two survey posts, \(860\) m apart. A ferry is at \(R\).

    The angle \(RPQ\) is \(59^\circ\) and the angle \(PQR\) is \(70^\circ\).

    860 mx mPQR59°70°Diagram NOT accurately drawn

    Find the distance \(QR\).

    Give your answer correct to the nearest metre.

    Show answer
    \(949\ \text{m}\)
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  6. Calculator 4 marks

    A triangular lawn \(PQR\) has \(QR = 25\) m and \(PQ = 22\) m.

    Angle \(QPR = 78^\circ\).

    (a)

    Find the size of angle \(PRQ\).

    Give your answer correct to 1 decimal place.

    (b)

    Find the size of angle \(PQR\).

    Give your answer correct to 1 decimal place.

    Show answer
    (a) \(59.4^\circ\)
    (b) \(42.6^\circ\)
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  7. Calculator 6 marks

    A boat sails \(47\) km north from \(A\) to \(B\).

    From \(B\) the boat sails to \(C\) and then back to \(A\).

    \(BC = 70\) km and the bearing of \(C\) from \(B\) is \(040^\circ\).

    N040°47 km70 kmABC
    (a)

    Show that the distance the boat sails from \(C\) to \(A\) is \(110\) km to the nearest km.

    You must show your working.

    (b)

    Work out the bearing of \(A\) from \(C\).

    Use your unrounded value of \(CA\) from part (a), not the value to the nearest km.

    Give your answer to the nearest degree.

    Show answer
    (a) Angle \(ABC = 140^\circ\), so \(CA^2 \approx 12149.57\) and \(CA \approx 110.23\) km \(\approx 110\) km
    (b) \(204^\circ\)
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  8. Calculator 5 marks

    \(ABC\) is a triangle.

    8 cmABC59°67°Diagram NOT accurately drawn
    (a)

    Find the size of the angle at \(C\).

    (b)

    Find the perimeter of the triangle.

    Give your answer to 1 decimal place.

    Show answer
    (a) \(54^\circ\)
    (b) \(25.6\ \text{cm}\)
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  9. Calculator 6 marks

    \(ABC\) is a triangle.

    8 cm7 cmABC82°Diagram NOT accurately drawn
    (a)

    Work out the length of \(AC\).

    Give your answer correct to 1 decimal place.

    (b)

    Work out the size of angle \(BAC\).

    Give your answer correct to the nearest degree.

    Show answer
    (a) \(9.9\ \text{cm}\)
    (b) \(45^\circ\)
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  10. Calculator 4 marks

    In triangle \(ABC\), \(BC = 7\) cm, \(AB = 12\) cm and angle \(BAC = 30^\circ\).

    Angle \(ACB\) is obtuse.

    Work out the size of angle \(ACB\).

    Give your answer to 1 decimal place.

    Show answer
    \(121.0^\circ\)
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discia.org The Sine Rule · sheet 1
Answers The Sine Rule · sheet 1
  1. \(\frac{ x }{ \sin 62^\circ } = \frac{ 16 }{ \sin 51^\circ }\)
  2. \(7.55\ \text{cm}\)
  3. \(53.8^\circ\)
  4. (a) \(97^\circ\)
    (b) \(5.1\ \text{cm}\)
  5. \(949\ \text{m}\)
  6. (a) \(59.4^\circ\)
    (b) \(42.6^\circ\)
  7. (a) Angle \(ABC = 140^\circ\), so \(CA^2 \approx 12149.57\) and \(CA \approx 110.23\) km \(\approx 110\) km
    (b) \(204^\circ\)
  8. (a) \(54^\circ\)
    (b) \(25.6\ \text{cm}\)
  9. (a) \(9.9\ \text{cm}\)
    (b) \(45^\circ\)
  10. \(121.0^\circ\)
discia.org The Sine Rule · sheet 1