Trigonometry: Angles of elevation and depression

Now that you've learnt about the triangular function Sine, we'll learn about another trigonometric functions, cosine, and its values. Cosine is also written as cos.

Angle of Elevation

If a person needs to look up past his horizontal line of sight in order to see an object, the angle created
here is the angle of elevation. The image above shows the angle of elevation created when a bug looks up
at the man above.

For example:The angle of elevation of the top of a flag pole measures 46° from a point on the ground 25 ft
away from its base. What is the height of the flagpole?

Answer: 26 ft

Tan 46° = x / 25 ft1.04 x 25 = xx = 26 ft

Angle of depression

If a person needs to look down past his horizontal line of sight in order to see an object, the angle created
here is the angle of depression. The image above shows the angle of depression when the man looks down
at the bug.For example:Han stood at the edge of a 40 m building. Using his binoculars, he looked down to
see his car at an angle that measures 38°. How far is his car from building?

Answer: 51.3 m

tan 38° = 40 / x

x = 40 / 0.78

x = 51.3 m

Tips on solving trigonometry problems:

• Show the values of the other angles and the lengths known

• Mark the angles and sides that need solving.

• Decide on the formula you will need to solve the unknown. Consider Pythagoras theorem, sine, cosine or tangent.

• Check that your answer is rational. The hypotenuse is always the longest side in any right triangle.

Try these questions:

1. Jessica stands on the top of a 130 m cliff and looks down at an angle of 46° at the yacht docked on the port.
   (a) How far is the object from the base of the cliff? (b) What is the angle of elevation created should any crew
   from the yacht sees Jessica on the top?

   Answer: (a) 125 m (b) 46°

   Explanation:

   
   

   

   
   

   a) tan 46° = 130 / x

   x = 130 / 1.04

   x = 125 m

   
   

   b) 90° - 46° = 44°

   The angle found inside the triangle is 44°.

   Therefore,

   180° = 90 + 44 + x

   180 – 90 – 44 = x

   46° = x




2. Ken is standing on the roof top of a 30 m building when he spots Leo, also on the roof top, of a 43 m building,
   puffing a smoke about 8m away. At what angle was Ken looking up at Leo? Sketch a diagram to represent the
   situation.

   Answer: 58°

   Explanation:

   
   

   

   
   

   Length of the opposite side of the triangle:

   43 m – 30 m = 13 m

   Therefore,

   tan x = 13 m / 8 m

   tan- 1.6 = x

   x = 58°




3. Eric who is 2 m tall looks up at a tree 30 m away from him. He plans to climb it and get the fruits before the birds can get to them. The elevation of the top of the tree from his eyes is 35˚. Estimate the height of the tree.

   Answer:

   Explanation:

   

   
   

   tan 35° = (x + 2m) / 30m

   0.7 * 30 - 2 = x

   19m = x