WAEC 2010 · Paper 2 · Q3✱

(a)
In a right-angled triangle, $\sin x = \frac35$ . Evaluate $5\cos^2 x - 3$ .
(b)
The angle of elevation of the top of a vertical pole from a point $63\text{ m}$ east of the base of the pole is $30^\circ$ . From another point due west of the pole, the angle of elevation of the top is $60^\circ$ . (i) Draw a sketch diagram to illustrate the information. (ii) Calculate, correct to three significant figures, the distance of the second point from the base of the pole.
Model answer
The pole $TB$ stands between the two points: $A$ is $63\text{ m}$ east of $B$ , $C$ is $x\text{ m}$ west of $B$ .

Worked solution (try it first)

Sketch the pole $TB$ with $A$ $63\text{ m}$ east of the foot $B$ (elevation $30^\circ$ ) and $C$ west of $B$ (elevation $60^\circ$ ), both on level ground.

From $A$ : the height is $|TB| = 63 \tan 30^\circ = 36.37\text{ m}$ .
From $C$ : $\tan 60^\circ = \dfrac{|TB|}{x}$ , so $x = \dfrac{63 \tan 30^\circ}{\tan 60^\circ}$ .
$\tan 30^\circ = \frac{1}{\sqrt3}$ and $\tan 60^\circ = \sqrt3$ , so $x = \frac{63}{3} = 21.0$ .
The second point is $21.0\text{ m}$ from the base of the pole.