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 ${45}^o$. The angle of inclination of the slope with the horizontal is

• A. ${60}^o$
• B. ${45}^o$
• C. ${75}^o$
• D. ${30}^o$

Answer 1

The correct option is C ${75}^o$

Applying m-n theorem of trigonometry, we get

$(c+c)\cot (\theta -30)=c\cot 15-c\cot 30$

$\Rightarrow \cot (\theta -30)=\frac{1}{2}\frac{\sin (30-15)}{\sin 15\sin 30}$

$\Rightarrow \cot (\theta -30)=\frac{1}{2}\frac{1}{\sin 30}$

$\Rightarrow \cot (\theta -30)=1$

$\Rightarrow \cot (\theta -30)=\cot 45$

$\Rightarrow \theta -30=45$

$\therefore \theta ={75}^o$

So, angle of inclination if the slope with horizontal will be ${75}^o$