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The Laws of Sines and Cosines · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SLaw of Sines and its applicationss5 marksPaper 1~8 min
A surveyor measures the angle of elevation to the top of a tower from point PP as 35°35°. From point QQ, which is 50 m50\,\text{m} closer to the tower along the same horizontal ground, the angle of elevation is 52°52°. Let hh be the height of the tower and let xx be the horizontal distance from QQ to the base of the tower.
(a)
Show that x=50tan⁡35°tan⁡52°−tan⁡35°x = \dfrac{50\tan 35°}{\tan 52° - \tan 35°}. [2 marks]
(b)
Find the height of the tower, correct to 3 significant figures. [3 marks]
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Solutions

2FoundationSAQ-SLaw of Sines and its applicationss5 marksPaper 1~8 min
A triangular garden has sides of length 14 m14\,\text{m} and 9 m9\,\text{m}, and the angle between these sides is 115°115°.
(a)
Show that the length of the third side is 19.6 m19.6\,\text{m}, correct to 3 significant figures. [2 marks]
(b)
Find the smallest angle of the triangle, correct to the nearest degree. [2 marks]
(c)
A fence is to be built along all three sides of the garden. Fencing costs USD 35 per metre. Determine the total cost of the fence, giving your answer correct to the nearest dollar. [1 mark]
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Solutions

3MasterySAQ-SLaw of Sines and its applicationss4 marksPaper 1~6 min
In triangle PQRPQR, PQ=12 cmPQ = 12\,\text{cm}, ∠PQR=42∘\angle PQR = 42^\circ, and ∠PRQ=73∘\angle PRQ = 73^\circ.
(a)
Show that QRsin⁡65∘=12sin⁡73∘\dfrac{QR}{\sin 65^\circ} = \dfrac{12}{\sin 73^\circ}. [2 marks]
(b)
Hence find the length of QRQR, giving your answer correct to 3 significant figures. [2 marks]
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Solutions

4ChallengeSAQ-LLaw of Sines and its applicationss8 marksPaper 1~12 min
A surveyor is measuring a triangular plot of land ABCABC. The distance AB=120 mAB = 120\,\text{m}, the distance AC=150 mAC = 150\,\text{m}, and the angle BA^C=40°B\hat{A}C = 40°.
(a)
Determine the length BCBC, giving your answer correct to 3 significant figures. [3 marks]
(b)
A path from AA meets BCBC at point DD such that ADAD bisects angle BA^CB\hat{A}C. State the angle bisector theorem and hence determine the exact ratio BD:DCBD : DC. [2 marks]
(c)
Find the length BDBD, giving your answer correct to 3 significant figures. [2 marks]
(d)
The surveyor claims that point DD lies closer to BB than to the midpoint MM of BCBC. Determine whether this claim is correct, justifying your answer with calculations. [1 mark]
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Solutions

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