Hi there. In this math post, I cover the topics of angles of elevation and angles of depression.
Drawings were done with the MathisFun drawing website and I took screenshots of them.
Topics
- Review Of Trigonometry Of Right Angled Triangles
- Angle Of Elevation
- Angle Of Depression
Review Of Trigonometry Of Right Angled Triangles
Before getting into the concepts of angles of elevation and angles of depression, I do think it is important to cover trigonometry and right angled triangles. In a right angled triangle we have three sides and three angles. One of the three sides is the longest side called the hypotenuse. This hypotenuse is opposite to the largest angle which is the right angle inside the triangle.
Screenshot picture from the Mathisfun.com website.
Given an angle such as theta θ we have the sides of the triangle related to where the angle θ is.
Trigonometry Ratios - Sine, Cosine & Tangent
The three basic trigonometric functions are sine (sin), cosine (cos) and tangent (tan). These functions are typically expressed as a fraction with the side lengths.
A popular memory aid to memorize these three fractions is SOH CAH TOA. SOH represents the sine of angle as the opposite side length divided by the hypotenuse side length. COH is adjacent side length divided by the hypotenuse and TOA is for the ratio of the opposite side length divided by the adjacent side length.
Angle Of Elevation
An angle of elevation is easy to visualize. Consider a ramp. The angle from the base of the ramp that touches the floor to the angled surface is the angle of elevation.
Example One
You have a ramp with an incline of 10 degrees as the angle of elevation. The bottom of the ramp is 8 metres long. What is the height of the ramp?
A picture helps with visualizing the information that is available and what is the unknown value in the right angled triangle.
There is the 8 metres for the bottom of the ramp along with the angle of elevation of 10 degrees. This 8 metres is the adjacent side to the 10 degrees. The height of the ramp is the opposite side to the 10 degrees.
The appropriate trigonometric ratio here is tangent.
Multiplying both sides by 8 gives the value for the height of the ramp.
Example Two
Johnny is looking at a mountain top that is 1400 metres away from him as the line of sight. The distance from Johnny's eyes to the top of the mountain is about 500 metres. At what angle should Johnny should tilt his head up from a straight head position?
Drawing done with Mathisfun Drawing website. That is why it looks as it is.
What is given is the hypotenuse of 1400 metres and the height of 500 metres. The height of the mountain is opposite to the angle of elevation. Set up the sine trigonometric ratio to start.
When it comes to solving for an unknown angle, use the appropriate inverse trigonometric function. In this case use the sine inverse function which looks like $\sin^{-1}$. On a scientific calculator you would need the SHIFT key, a second function or something similar to find a trig inverse function.
Angle Of Depression
The angle of depression is an angle that measures the distance between two lines where one is horizontal and the second line dips down away from the horizontal line.
From this website, here is a nice summary picture for the angle of elevation and the angle of depression.
Example - Annoying Laser Pointing Guy At Soccer Stadium
This laser pointing guy wants to point his laser at some goalie's eyes as a distraction in a free kick scenario. This guy is about 100 metres away from the goalie diagonally as a line of sight. The angle of depression from this guy is 30 degrees. What is the height difference of where this laser is to the goalie's eyes?
The available information includes the hypotenuse as 100 metres and the angle of depression of 30 degrees. If you draw the triangle a different way the 30 degree angle of depression would not be shown.
Set up the sine trigonometric ratio and then solve for the height (h).
The height difference of the laser is to the goalie's eyes is 50 metres.
