# 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.

^{o}

Answer

`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`.

Answer

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* = 90^{o}.

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

So *a* = 15.67 sin 35^{o} = 8.99

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

So *b* = 15.67 cos 35^{o} = 12.84

Angle *B *= 90^{o} − 35^{o} = 55^{o}.

So *a* = 8.99, *b* = 12.84 and *B* = 55^{o}. We have found all the unknowns.

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## 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?

Answer

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

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

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### 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?

Answer

This is a simple tan ratio problem.

`tan 8^"o" = 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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