Solve the whole triangle.

The cosine rule relates two sides and their included angle to the opposite side. It reduces to Pythagoras when the included angle is 90°. The sine rule pairs each side with the sine of its opposite angle.

Given two sides and a non-included angle, the sine rule can produce zero, one or two triangles. If $\sin B$ = s, consider both B = arcsin(s) and 180° − arcsin(s), then reject candidates whose angle sum is at least 180°. Draw bearings and elevations before translating them into triangle angles.

INTERACTIVE MODEL

Try it. Watch it change.

Side a control (a = 4 + control)1Side b control (b = 5 + control)0Angle C / degrees78

c² = a² + b² − 2ab cos C · area = 12.2268 square units

Sides and angle are calculated to scale. At 90°, the cosine term is zero and c is the hypotenuse.

Explore: Change the included angle while holding both sides fixed. Compare area and the opposite side.

Cosine rule

c^2=a^2+b^2-2ab\cos C

Triangle area

A=\frac12 ab\sin C

WORKED EXAMPLE

Two sides are 5 m and 8 m with included angle 60°. Find the opposite side.

$c^{2}$ = $5^{2}$ + $8^{2}$ − 2(5)(8)cos 60° = 49.
c = 7 m.

Assumed knowledge

Trigonometric ratios and angle sums.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use sine and cosine rules in non-right triangles.
Check the ambiguous sine-rule case and calculate area.

QCAA Mathematical Methods 2025 v1.3 · p. 28 ↗

READY TO TRY IT?

Solve the whole triangle.

Two sides are 5 m and 8 m with included angle 60°. Find the opposite side.

Put the idea to work.