Application to Angle of Elevation and Depression - JSS3 Mathematics Lesson Note

Angle of Elevation:

Definition: Angle of elevation is the angle formed when an observer looks up from horizontal to see an object above the horizontal level.

Application: To find the angle of elevation to an object, trigonometric ratios are used based on the height of the object and the horizontal distance from the observer.

Example: Suppose an observer is looking at the top of a building. The observer stands 50 meters away from the base of the building and looks up at a 30-degree angle to see the top of the building. We can use the tangent ratio to find the height of the building:

Given:

Horizontal distance (adjacent side, adj) = 50 meters

Angle of elevation (θ) = 30 degrees

To find the height (opposite side, opp):

tan(30∘)=opp50=50⋅tan(30 ∘ )

opp≈50⋅0.577=28.85 meters

So, the height of the building is approximately 28.85 meters.

Angle of Depression:

Definition: Angle of depression is the angle formed when an observer looks down from horizontal to see an object below the horizontal level.

Application: Similarly, trigonometric ratios are used based on the known measurements to find the angle of depression.

Example: If an observer looks down at an angle of 20 degrees to see the top of a flagpole directly below their position, and they are standing 30 meters above the ground, we can find the height of the flagpole using the tangent ratio:

Given:

Height above ground (opposite side, opp) = 30 meters

Angle of depression (θ) = 20 degrees

To find the height of the flagpole (adjacent side, adj):

tan⁡(20∘)=30adj

adj=30tan(20∘)

adj≈ 0.364/30=82.42 meters

So, the height of the flagpole is approximately 82.42 meters.