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

1FoundationSAQ-SLaw of Sines and its applications7 marksPaper 1~11 min
A surveyor needs to find the distance across a small lake. She identifies two points, AA and BB, on opposite sides of the lake. From a third point CC on the shore, she measures AC=120 mAC = 120\,\text{m}, BC=95 mBC = 95\,\text{m}, and AC^B=52°A\hat{C}B = 52°.
(a)
On the below, label the sides opposite to vertices AA, BB, and CC as aa, bb, and cc respectively, and write down the Law of Cosines in the form c2=…c^2 = \ldots [1 mark]
(b)
Calculate the distance ABAB, giving your answer correct to 3 significant figures. [3 marks]
(c)
The surveyor claims that if the angle AC^BA\hat{C}B were increased to 90°90° while ACAC and BCBC remained unchanged, the distance ABAB would increase by more than 10 m10\,\text{m}. Determine whether this claim is correct, justifying your answer. (: triangle with vertices labelled AA, BB, CC; no side or angle values marked.) [3 marks]
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Solutions

2MasterySAQ-SLaw of Sines and its applications8 marksPaper 1~12 min
In triangle ABCABC, side a=8.0 cma = 8.0\,\text{cm}, side b=10.0 cmb = 10.0\,\text{cm}, and angle A=35°A = 35°.
(a)
Show that sin⁡B=0.717\sin B = 0.717, correct to three significant figures. [3 marks]
(b)
Hence write down the two possible values of angle BB, correct to the nearest degree. [1 mark]
(c)
Determine which value(s) of BB give a valid triangle. For each valid triangle, find the corresponding length of side cc, correct to one decimal place. [4 marks]
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Solutions

3ChallengeSAQ-LLaw of Sines and its applications8 marksPaper 1~12 min
A surveyor is mapping a triangular parcel of land ABCABC. The distance AB=120 mAB = 120\,\text{m}, angle A^=52°\hat{A} = 52°, angle B^=63°\hat{B} = 63°, and BC=95 mBC = 95\,\text{m}.
(a)
Determine the length of ACAC, correct to the nearest metre. [4 marks]
(b)
Point DD lies on BCBC such that ADAD bisects angle A^\hat{A}. Using the angle bisector theorem and the cosine rule, determine the length of ADAD, correct to 3 significant figures. [4 marks]
diagram

Solutions

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