MATH / IGCSE

Trigonometry and bearings

Sine, cosine and tangent connect right-triangle side ratios with angles. The sine and cosine rules extend calculations to general triangles.

Learning objectives

Explain trigonometry and bearings using a concrete example.
Apply the worked method and justify each step.
Recognise the limitation described in the final check.

The big idea

Sine, cosine and tangent connect right-triangle side ratios with angles. The sine and cosine rules extend calculations to general triangles.

A bearing is measured clockwise from north using three figures. Draw a labelled diagram before choosing a rule; distinguish a horizontal distance from the sloping side and set the calculator angle mode.

Equations and relationships

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

Right-triangle ratio.

Worked example

A right triangle has hypotenuse 10 cm and angle 30°. Find its opposite side.

Follow the solution

opposite = 10 sin 30° = 10 × 0.5 = 5 cm.

Investigate and explain

Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.

Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.

Open the interactive investigation

Check your understanding

1. What bearing points east?

000°
180°
090°

Check the answer and reasoning

090°

Bearings start at north and increase clockwise.

2. A right triangle has hypotenuse 10 cm and angle 30°. Find its opposite side.

5 cm
20 cm
8.66 cm

Check the answer and reasoning

5 cm

opposite = 10 sin 30° = 10 × 0.5 = 5 cm.

3. Which rule finds a third side from two sides and their included angle?

Pythagoras for every angle
Cosine rule
Sine rule without another angle

Check the answer and reasoning

Cosine rule

The cosine rule uses a² = b²+c²−2bc cos A and does not require a right angle.