Theory question 107
Trigonometry(a) A woman looking out from the window of a building at a height of 30 m, observed that the angle of depression of the top of a flag pole was 44°. If the foot of the pole is 25 m from the foot of the building and on the same horizontal ground, find, correct to the nearest whole number, the (i) angle of depression of the foot of the pole from the woman; (ii) height of the flag pole.
(b) In the diagram, O is the centre of the circle, ∠OQR = 32° and ∠TPQ = 15°. Calculate: (i) ∠QPR; (ii) ∠TQO.
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(a)(i) tan θ = 30/25 = 1.2 → θ ≈ 50°.
(ii) Drop from the window to the level of the pole's top: 30 − h = 25 tan 44° ≈ 24.14, so h ≈ 5.86 ≈ 6 m.
(b)(i) OQ = OR, so ∠ORQ = 32° and ∠QOR = 180° − 64° = 116°. ∠QPR = ½ × 116° = 58°.
(ii) ∠TOQ = 2 × ∠TPQ = 30°. OT = OQ, so ∠TQO = (180° − 30°)/2 = 75°.
Radii make isosceles triangles; the angle at the centre is twice the angle at the circumference.