Module 9: Geometry

An angle of elevation is any angle above a horizontal plane. For example, imagine you are standing 10 m away from the base of an apartment complex. You look up along the side of the building and notice a balcony 25 m up from your horizontal. The angle of elevation here is the angle you have to look up at to observe said balcony relative to your horizontal. The diagram summarizes this scenario.

What if we wanted to solve for the angle of elevation? At what angle are you looking upwards at? To solve this, consider what information we already know and what trigonometric method/ratio can be used given what is known. Since the opposite and adjacent side lengths are known, the tangent ratio can be used to solve for the angle of elevation.

$${"version":"1.1","math":"tan\;\theta=\dfrac{opposite}{adjacent}"}$$

$${"version":"1.1","math":"\theta=tan^{-1}\dfrac{opposite}{adjacent}"}$$

$${"version":"1.1","math":"\theta=tan^{-1}\dfrac{25}{10}"}$$

$${"version":"1.1","math":"\theta\cong68.2^\circ"}$$

An angle of depression is any angle below a horizontal plane. For example, imagine you are standing on a 25m tall cliff overlooking water and spot a boat in the water 10 m away from the base of the cliff. Let’s say we wanted to find the distance between the top of the cliff and the boat. A triangle can be formed based off this information, as modelled below:

Here, angle ${"version":"1.1","math":"\theta"}$ is the angle of depression. A triangle can be formed along the cliffside. If we know the angle of depression, we can determine the angle at the top of the triangle (${"version":"1.1","math":"a"}$) by subtracting ${"version":"1.1","math":"\theta"}$ from 90° (which is the angle between the cliffside and horiontal). Alternatively, a triangle can be made that is upside-down, as shown by the dotted triangle. The angles are reflected at the bottom. It is up to you which you prefer to use.

To find the distance between the top of the cliff and the boat, we’ll need to find the length of the hypotenuse. Since we are only given two side lengths and need to find the third, we can use Pythagorean theorem to solve for the hypotenuse. If the angle of depression was given, we can make use of trigonometric ratios or quickly verify if the special cases apply to solve this problem instead (whichever ratio is chosen depends on your diagram and which pieces of information you want to use).

$${"version":"1.1","math":"a^2+b^2=c^2"}$$

$${"version":"1.1","math":"c=\sqrt{a^2+b^2}"}$$

$${"version":"1.1","math":"c=\sqrt{10^2+25^2}"}$$

$${"version":"1.1","math":"c=\sqrt{100+625}"}$$

$${"version":"1.1","math":"c=\sqrt{725}"}$$

$${"version":"1.1","math":"c\cong26.9"}$$

In any case, all triangles that are formed will be right triangles, thus SOH CAH TOA, Pythagorean theorem, and angle calculations can all be used to solve these types of problems.