Question

A 6 feet tall man finds an apple on the ground 6$\sqrt{3}$ feet away from him. What is the angle of depression when he is looking at the apple (in degrees)? ___

Answer 1

Imagine the man standing to be a vertical straight line, AB, and the position of the apple to be at a point C.

Now if we connect the man’s head and feet to the position of the apple we can imagine it to be a right triangle with vertices ABC.

We already know that the height of the man is 6 feet and he is standing 6$\sqrt{3}$ feet away from the apple. Therefore, we can say that, AB = 6 feet and BC = 6$\sqrt{3}$ feet.

The angle of depression is the angle the line of sight makes with the horizontal level, i.e. the angle between AC and AD. Let us name the angle θ.

$\angle$ DAB = $\angle$ CBA = ${90}^{\circ }$.

Therefore we can say that DA $\parallel$ BC.

Now we can say that $\angle$ DAC = $\angle$ ACB, because they are alternate interior angles of the parallel lines AD and BC.

Let’s apply trigonometric ratios in the right triangle $△$ ABC to find $\theta$

$\tan$ $\theta$ = $\frac{AB}{BC}$

= $\frac{1}{\sqrt{3}}$

= $\theta$ = ${30}^{\circ }$ $\because tan\left({30}^{\circ }\right)=\frac{1}{\sqrt{3}}$