Question

A man observes that when he moves up a distance $c$ metres on a slope, the angle of depression of a point on the horizontal plane from the base of the slope is ${30}^o$; and when he moves up further a distance $c$ metres the angle of depression of that point is ${45}^o$. Obtain the angle of inclination of the slope with the horizontal.

Answer 1

Applying $m-n$ theorem of trigonometry, we get
[$(\theta -{30}^o)$ on the left]
$-(c+c)\cot (\theta -{30}^o)$
$=c\cot {30}^o-c\cot {15}^o$
or $\cot (\theta -{30}^o)=\frac{1}{2}\frac{\sin ({30}^o-{15}^o}{\sin {15}^o\sin {30}^o}$
or $\cot (\theta -{30}^o)=\frac{1}{2}\frac{1}{\sin {30}^o}$
$\therefore \theta ={30}^o={45}^o$ or $\theta ={75}^o$