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

1FoundationSAQ-SLaw of Sines and its applicationss5 marksPaper 1~8 min
A surveyor needs to find the height of a vertical cliff. From a point PP on the ground, the angle of elevation to the top of the cliff is 30∘30^\circ. The surveyor then walks 100 m100\,\text{m} directly towards the cliff to a point QQ. From QQ, the angle of elevation to the top of the cliff is 45∘45^\circ. Let the height of the cliff be hh metres and let the horizontal distance from PP to the base of the cliff be dPd_P metres.
(a)
Show that dP=h3d_P = h\sqrt{3}. [2 marks]
(b)
Write down the horizontal distance from QQ to the base of the cliff in terms of hh. [1 mark]
(c)
Hence, find the value of hh, giving your answer in the form a+b3a + b\sqrt{3} where a,b∈Za, b \in \mathbb{Z}. [2 marks]
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Solutions

2MasterySAQ-SLaw of Sines and its applicationss5 marksPaper 1~8 min
In triangle PQRPQR, PQ=10 cmPQ = 10\,\text{cm}, ∠PQR=42∘\angle PQR = 42^\circ, and ∠PRQ=68∘\angle PRQ = 68^\circ.
(a)
Show that PR=10sin⁡42∘sin⁡68∘PR = \dfrac{10\sin 42^\circ}{\sin 68^\circ}. [2 marks]
(b)
Using the identity sin⁡68∘=cos⁡22∘\sin 68^\circ = \cos 22^\circ and the result from part (a), show that PR=10sin⁡42∘cos⁡22∘PR = \dfrac{10\sin 42^\circ}{\cos 22^\circ}. [1 mark]
(c)
Hence, using an appropriate double-angle identity, show that PR=5sin⁡42∘sin⁡11∘cos⁡11∘PR = \dfrac{5\sin 42^\circ}{\sin 11^\circ \cos 11^\circ}, and deduce an expression for PRPR in the form ksin⁡42∘sin⁡22∘\dfrac{k\sin 42^\circ}{\sin 22^\circ}, stating the value of kk. [2 marks]
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Solutions

3ChallengeSAQ-LLaw of Sines and its applicationss8 marksPaper 1~12 min
In triangle ABCABC, side AB=10 cmAB = 10\,\text{cm}, side BC=8 cmBC = 8\,\text{cm}, and ∠BAC=θ\angle BAC = \theta, where θ\theta is acute and sin⁡θ=45\sin\theta = \dfrac{4}{5}.
(a)
Determine the two possible values of ∠ACB\angle ACB, giving your answers in degrees correct to one decimal place. [4 marks]
(b)
For each value of ∠ACB\angle ACB found in part (a), determine the corresponding length of ACAC, giving your answers in cm correct to three significant figures. [2 marks]
(c)
Analyse whether the two possible triangles have the same area. Support your answer with appropriate calculations. [2 marks]
diagram

Solutions

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