4. The Right Triangle and Applications

Many problems involve right triangles. We often need to use the trigonometric ratios to solve such problems.

Example 1 - Finding the Height

Find h.

math expression

Answer: sin15.7° =

h = 2.500 sin 15.7°

= 0.6765 km

Example 2 - Solving Triangles

Solve the triangle ABC, given that A = 35° and c = 15.67.

Answer: To "solve" a triangle means to find the unknown sides and angles. In this example, we need to find a and b and angle B. Note C = 90°.

We have

So a = 15.67 sin 35° = 8.99

So b = 15.67 cos 35° = 12.84

Angle B = 90° - 35° = 55°.

So a = 8.99, b = 12.84 and B = 55°. We have found all the unknowns.

Angles of Elevation and Depression

In surveying, the angle of elevation is the angle from the horizontal looking up to some object:

math expression

The angle of depression is the angle from the horizontal looking down to some object:

math expression

Example 3:

The angle of elevation of an aeroplane is 23°. If the aeroplane's altitude is 2500 m, how far away is it?

math expression

Answer: Let the distance be x. Then .

Example 4

You can walk across the Sydney Harbour Bridge and take a photo of the Opera House from about the same height as top of the highest sail.

opera house

This photo was taken from a point about 500 m horizontally from the Opera House and we observe the waterline below the highest sail as having an angle of depression of 8°. How high above sea level is the highest sail of the Opera House?

Answer: This is a simple tan ratio problem.

tan 8° = h/500

So

h = 500 tan 8° = 70.27 m.

So the height of the tallest point is around 70 m.

[The actual height is 67.4 m.]

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