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Chapter Contents

• Trigonometric Functions
• 1. Angles
• 2. Sine, Cosine, Tangent & Reciprocals
• 3. Values of Trigonometric Functions
• 4. The Right Triangle and Applications
• 5. Signs of the Trigonometric Functions
• 6. Trigonometric Functions of Any Angle
• 7. Radians
• 8. Applications of Radian Measure
• Pulleys
• 9. Radians and the Trigonometric Ratios

• Comments, Questions?

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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.

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:

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 .

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° = 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.]