Mathematics

Mathematics

Problem

A vertical pole stands on level ground. From a point PP on the ground, the angle of elevation of the top of the pole is θ\theta such that tan⁡θ=512\tan\theta = \dfrac{5}{12}. Using this information, answer the following. a. (i) Calculate sec⁡θ\sec\theta. b. (ii) Determine sin⁡θ\sin\theta in exact fractional form. c. (iii) If the distance  PQ \,PQ\, from the point PP to the foot QQ of the pole is 36 m36\,\text{m}, find the height of the pole to the nearest centimetre.
Intuitive explanation

The key is recognizing that tan⁡θ=512\tan\theta = \frac{5}{12} encodes a right triangle hidden in the geometry—the one formed by the pole (vertical), the ground distance from PP (horizontal), and the line of sight to the top. Once you sketch this triangle and label it with the ratio, you can extract any other measurement you need using the Pythagorean theorem and basic trigonometry.

Intuitive explanation

The key is recognizing that tan⁡θ=512\tan\theta = \frac{5}{12} encodes a right triangle hidden in the geometry—the one formed by the pole (vertical), the ground distance from PP (horizontal), and the line of sight to the top. Once you sketch this triangle and label it with the ratio, you can extract any other measurement you need using the Pythagorean theorem and basic trigonometry.

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