36 exam-style questions on Applications of trig; elevation/depression; labelled diagrams — topic 3.3 of Geometry and Trigonometry in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
A surveyor stands 50 m from the base of a vertical tower and measures the angle of elevation to the top as 32°. Calculate the height of the tower to 3 s.f.
Answer: 31.2 m
height = 50·tan32° = 50(0.6249) = 31.2 m. Using sin32° instead of tan32° gives the incorrect 26.5 m.
A ramp rises 1.5 m over a horizontal run of 6 m. Calculate the angle of elevation of the ramp to the nearest degree.
Answer: 14°
tan θ = 1.5/6 = 0.25, so θ = tan⁻¹(0.25) ≈ 14°. Answer 76° is the complement, from dividing 6/1.5 by mistake.
A ship sails on a bearing of 120° for 30 km. Calculate how far east of its start point it now lies, to 3 s.f.
Answer: 26.0 km
A bearing of 120° is 30° south of due east, so the east component = 30·cos30° = 30·sin60° = 30(0.8660) = 26.0 km.
An aeroplane flies at a constant altitude of 2000 m. From a point on the ground the angle of elevation to the plane is 25°. Calculate the horizontal distance from that point to the point directly below the plane, to 3 s.f.
Answer: 4290 m
tan25° = 2000/distance, so distance = 2000/tan25° = 2000/0.4663 = 4290 m.
A hiker walks 5 km on a bearing of 060°, then 4 km on a bearing of 150°. The angle between the two legs of the path (the interior angle at the turning point) is 90°. Calculate the straight-line distance back to the start, to 3 s.f.
Answer: 6.40 km
The two bearings differ by 90°, so the legs are perpendicular; by Pythagoras the distance = √(5² + 4²) = √41 = 6.40 km.
A triangular plot has two sides of 80 m and 65 m meeting at an angle of 70°. Calculate the area of the plot to 3 s.f.
Answer: 2440 m²
Area = ½(80)(65)sin70° = 2600·(0.9397) = 2440 m². Omitting the sine factor gives the incorrect 2600 m².
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36 Applications of trig; elevation/depression; labelled diagrams questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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