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Trigonometric Ratios and Identities · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-SDefinitions of sine, cosine and tangent using right-angled triangles5 marksPaper 1~8 min
A ladder leans against a vertical wall. The foot of the ladder is 1.5 m1.5\,\text{m} from the base of the wall. The ladder makes angle of 72°72° with the horizontal ground.
(a)
Write an equation using the cosine ratio to express the relationship between the length of the ladder ll and the 1.5 m1.5\,\text{m} horizontal distance. [1 mark]
(b)
Find the length of the ladder ll. [2 marks]
(c)
The top of the ladder rests against the wall at height hh. Find hh, giving your answer correct to 3 significant figures. [2 marks]
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Solutions

2FoundationSAQ-SDefinitions of sine, cosine and tangent using right-angled triangles5 marksPaper 1~8 min
A surveyor measures the angle of elevation to the top of a tree as 35°35°. The surveyor stands 20 m20\,\text{m} from the base of the tree. The tree is vertical and the ground is horizontal.
(a)
Write down the trigonometric ratio that relates the height of the tree, the distance from the base, and the angle of elevation. [1 mark]
(b)
Find the height of the tree. [2 marks]
(c)
The surveyor moves closer to the tree until the angle of elevation becomes 50°50°. Find how much closer, in metres, the surveyor has moved. [2 marks]
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Solutions

3MasterySAQ-SDefinitions of sine, cosine and tangent using right-angled triangles5 marksPaper 1~8 min
A surveyor measures the angle of elevation to the top of a vertical cliff from a point on level ground as 35∘35^\circ. She then walks 5050 metres directly towards the cliff and measures the angle of elevation as 52∘52^\circ. Let the height of the cliff be hh metres and let the initial distance from the base of the cliff to the surveyor be dd metres.
(a)
Show that h=dtan⁡35∘h = d\tan 35^\circ. [1 mark]
(b)
Show that h=(d−50)tan⁡52∘h = (d - 50)\tan 52^\circ. [1 mark]
(c)
Hence find the height of the cliff. Give your answer correct to 3 significant figures. [3 marks]
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Solutions

4ChallengeSAQ-LDefinitions of sine, cosine and tangent using right-angled triangles8 marksPaper 1~12 min
A surveyor measures the height of a vertical cliff face. From point A, at ground level, the angle of elevation to the top of the cliff is 35∘35^\circ. The surveyor walks 120 m120\,\text{m} directly towards the cliff to point B, where the angle of elevation to the top is 52∘52^\circ. Let the height of the cliff be hh metres and the horizontal distance from B to the base of the cliff be xx metres.
(a)
Show that h=(x+120)tan⁡35∘h = (x + 120)\tan 35^\circ and h=xtan⁡52∘h = x\tan 52^\circ. [2 marks]
(b)
Hence, determine the value of xx, correct to 3 significant figures. [3 marks]
(c)
Hence, determine the height hh, correct to 3 significant figures. [1 mark]
(d)
The surveyor's clinometer has a measurement uncertainty of ±1∘\pm 1^\circ. Using 34∘34^\circ instead of 35∘35^\circ for the angle at A gives h=171 mh = 171\,\text{m} (3 s.f.). Evaluate whether the surveyor's measurement of hh is reliable given this uncertainty. [2 marks]
diagram

Solutions

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