In these lessons, we will study some practical applications of trigonometry in the calculation of angles of elevation and angles of depression.
The angle of elevation is the angle between a horizontal line from the observer and the line of sight to an object that is above the horizontal line.
In the diagram below, AB is the horizontal line. θ is the angle of elevation from the observer at A to the object at C.
The angle of depression is the angle between a horizontal line from the observer and the line of sight to an object that is below the horizontal line.
In the diagram below, PQ is the horizontal line. θ is the angle of depression from the observer at P to the object at R.
Find angles of depression and angles of elevation, and the relationship between them.
Angles of elevation and depression are equal.
In the diagram below, AB and CD are two vertical poles on horizontal ground. Draw in the angle of elevation of D from B and the angle of depression of C from B.
Step 1: Draw a sketch of the situation.
Step 2: Mark in the given angle of elevation or depression.
Step 3: Use trigonometry to find the required missing length.
Two poles on horizontal ground are 60 m apart. The shorter pole is 3 m high. The angle of depression of the top of the shorter pole from the top of the longer pole is 20˚. Sketch a diagram to represent the situation.
Step 1: Draw two vertical lines to represent the shorter pole and the longer pole.
Step 2: Draw a line from the top of the longer pole to the top of the shorter pole. (This is the line of sight).
Step 3: Draw a horizontal line to the top of the pole and mark in the angle of depression.
A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Estimate the height of the tree.
Let the height of the tree be h. Sketch a diagram to represent the situation.
tan 28˚ =
h – 2 = 30 tan 28˚
h = (30 ´ 0.5317) + 2 ← tan 28˚ = 0.5317
h = 17.95
The height of the tree is approximately 18.0 m. (rounded)
Applications of Trig Functions: Solving for unknown distances.
How far away is a boat from the lighthouse if the lighthouse is 120° tall and the angle of depression to boat is 15°?
A boat is 500 meters from the base of a cliff. Jackie, who is sitting in the boat, notices that the angle of elevation to the top of the cliff is 32°15'. How high is the cliff? (Give your answer to the nearest meter).
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