Define — 2 marks
A surveyor is measuring the height of a cliff by standing 50 m away from its base and measuring the angle of elevation to the top as 35°. To solve this problem accurately, the surveyor needs to understand the trigonometric ratios and how they relate to right-angled triangles.
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(a) Define the sine ratio in a right-angled triangle.
[1 mark]
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(b) In the surveyor's measurement, identify which trigonometric ratio (sine, cosine, or tangent) should be used to find the height of the cliff, and define why this ratio is appropriate for this situation.
[1 mark]
Show mark scheme
- (a) Sine is the ratio of the opposite side to the hypotenuse in a right-angled triangle / sin(θ) = opposite/hypotenuse
- (b) Tangent should be used because the angle of elevation (35°), the adjacent side (50 m horizontal distance), and the opposite side (height) form a right-angled triangle where tan(θ) = opposite/adjacent, allowing calculation of height without knowing the hypotenuse
Calculate — 2 marks
A surveyor is measuring the height of a vertical cliff face. She stands 45 m horizontally away from the base of the cliff and measures the angle of elevation to the top of the cliff as 32°.
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Calculate the height of the cliff. Give your answer to 2 significant figures.
[2 marks]
Show mark scheme
- Correct identification and use of trigonometric ratio: tan(32°) = height/45 or height = 45 × tan(32°)
- Correct calculation and answer: 28 m (or 28.1 m to 3 s.f., or 2.8 × 10¹ m in standard form)
Show — 3 marks
A student is setting up a ramp to help move a heavy box up onto a platform. The ramp is positioned so that the vertical height from the ground to the platform is 1.2 m, and the horizontal distance from the base of the platform to the bottom of the ramp is 0.9 m.
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(a) Show that the length of the ramp is 1.5 m. (You may use Pythagoras' theorem: a² + b² = c²)
[2 marks]
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(b) Calculate the angle that the ramp makes with the horizontal ground. (Give your answer to 1 decimal place.)
[1 mark]
Show mark scheme
- (a) Correct substitution into Pythagoras' theorem: 1.2² + 0.9² = c² (or equivalent showing 1.44 + 0.81 = c²) - 1 mark
- (a) Correct calculation and conclusion: c² = 2.25, therefore c = 1.5 m - 1 mark
- (b) Use of trigonometry (sin, cos or tan with correct angle identification) and correct answer of 53.1° (accept 53.13°) - 1 mark
Compare — 4 marks
A student is designing two different ramp systems for a skateboard park. Ramp A is 3 m long and rises vertically by 1.5 m. Ramp B is 4 m long and rises vertically by 2 m.
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(a) Calculate the horizontal distance covered by Ramp A using Pythagoras' theorem.
[1 mark]
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(b) Calculate the horizontal distance covered by Ramp B using Pythagoras' theorem.
[1 mark]
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(c) Compare the angles of inclination of both ramps using trigonometry. Which ramp is steeper? You must show your working.
[2 marks]
Show mark scheme
- (a) Horizontal distance = √(3² - 1.5²) = √(9 - 2.25) = √6.75 = 2.60 m (to 2 d.p.) [1 mark]
- (b) Horizontal distance = √(4² - 2²) = √(16 - 4) = √12 = 3.46 m (to 2 d.p.) [1 mark]
- (c) Ramp A: sin(θ) = 1.5/3 = 0.5, so θ = 30° [1 mark for correct method and calculation]
- (c) Ramp B: sin(θ) = 2/4 = 0.5, so θ = 30°. Both ramps have the same angle of inclination / are equally steep. [1 mark for correct comparison and conclusion]
Calculate — 2 marks
A rectangular flower bed measures 3.6 m by 4.8 m. A gardener wants to install a straight path diagonally across the flower bed.
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(a) Calculate the length of the diagonal path. Give your answer in metres.
[1 mark]
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(b) The gardener has enough paving slabs to make a path of length 7 metres. Show that this is enough for the diagonal path.
[1 mark]
Show mark scheme
- (a) √(3.6² + 4.8²) or equivalent correct method seen
- (a) 6 (m)
- (b) Comparison of their answer to part (a) with 7 m (or 7² = 49 with their 6² = 36)
- (b) Correct conclusion based on their working (e.g., 6 < 7, so yes)
GCSE Perfect