Trigonometry is used to find unknown sides or angles in right angled triangles. These methods are also applied to real life problems involving bearings and angles of elevation and depression.

 

 

Trigonometric Ratios

There are three basic trigonometric ratios used in right angled triangles.

 

They are based on the angle \( \theta \) and the sides relative to that angle.

 

$$
\sin \theta = \frac{opposite}{hypotenuse}
$$

$$
\cos \theta = \frac{adjacent}{hypotenuse}
$$

$$
\tan \theta = \frac{opposite}{adjacent}
$$

 

These ratios only apply to right angled triangles.

 

The hypotenuse is always the longest side and is opposite the right angle

 

 

Choosing the Correct Ratio

To decide which ratio to use, identify:
• the angle you are working from
• the sides you know
• the side you need to find

 

For example:
• opposite and hypotenuse use sine
• adjacent and hypotenuse use cosine
• opposite and adjacent use tangent

 

Writing down the ratio first helps avoid mistakes.

 

 

Finding a Missing Side

When an angle and one side are known, trigonometry can be used to find another side.

 

Example
A right angled triangle has an angle of
\( 30^\circ \)
and a hypotenuse of 10 cm.

 

To find the opposite side:
$$
\sin 30^\circ = \frac{opposite}{10}
$$

 

Rearranging:
$$
opposite = 10 \sin 30^\circ
$$

 

 

Finding a Missing Angle

When two sides are known, trigonometry can be used to find an angle.

 

Example
The opposite side is 5 cm and the adjacent side is 8 cm.

 

$$
\tan \theta = \frac{5}{8}
$$

$$
\theta = \tan^{-1}\left(\frac{5}{8}\right)
$$

 

Angles are found using the inverse trigonometric functions on a calculator.

 

 

Angles of Elevation and Depression

An angle of elevation is measured upwards from the horizontal.

 

An angle of depression is measured downwards from the horizontal.

 

These angles are equal because they are alternate angles formed by parallel horizontal lines.

 

In problems:
• draw a clear diagram
• identify the right angled triangle
• apply the correct trigonometric ratio

 

The horizontal line is always important in these questions

 

Example Context
A person looks up at the top of a building at an angle of elevation of
\( 25^\circ \).
The horizontal distance to the building is known.

 

Trigonometry is used to find the height of the building.

 

 

Bearings and Trigonometry

When trigonometry is used with bearings, the direction is given first, then distances are calculated.

 

Bearings are measured clockwise from north.

 

To solve these problems:
• draw a north line
• mark the bearing angle
• create a right angled triangle
• use sine, cosine or tangent as required

 

Trigonometry helps find distances between points when bearings and one length are known.

 

 

Common Errors to Avoid

Common mistakes include:
• using trigonometry in a non right angled triangle
• choosing the wrong ratio
• mixing up opposite and adjacent sides
• forgetting to draw a diagram

 

Clear diagrams reduce most errors

 

 

Key Points to Remember

Trigonometry only applies to right angled triangles.
SOH CAH TOA helps remember the ratios.
Inverse functions are used to find angles.
Angles of elevation and depression are measured from the horizontal.
Bearings problems often require trigonometry to find distances.

 

Using trigonometric relationships confidently allows a wide range of geometric and real life problems to be solved accurately.