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# 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 for the given triangle.

15.70o

Triangle with unknown height h.

sin 15.70^"o" = h/2.500

h = 2.500\ sin 15.70^"o"

= 0.6765\ "km"

### Example 2 - Solving Triangles

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

To "solve" a triangle means to find the unknown sides and angles.

We have, for the values given:

In this example, we need to find side lengths a and b and angle B. Note C = 90o.

sin 35^"o" = a/c = a/15.67

So a = 15.67 sin 35o = 8.99

cos 35^"o"=b/c=b/15.67

So b = 15.67 cos 35o = 12.84

Angle B = 90o − 35o = 55o.

So a = 8.99, b = 12.84 and B = 55o. 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:

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

### Example 3

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

Let the distance be x. Then sin 23^"o"=2500/x

x=2500/(sin 23^"o")=6400\ "m"

### 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.

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?

tan 8^"o" = h/500
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".]