Sine and cosine rule

Sine rule

Note: the angles are labelled with a capital letter and the sides are labelled with a lower-case letter. The diagram below shows the formulas that we need to calculate the missing angle or side using the sin rule.

Let’s try an example to calculate a missing angle. So, we have to use the (1)^(st) formula.

\frac { sin x }{ 4 } =\frac { sin 75 }{ 9 }

Rearrange the formula to have sin x on its own.

sin x =\frac { 4\quad \times sin 75 }{ 9 }

Sin x = 0.42930…

In the calculator, press sin\quad x^{ -1 }0.49230  to find x. x = 25.4°

Cosine rule

The cosine rule can also be used to calculate missing lengths and angles.

Below shows the formulas for the cosine rule. The ones on the left are used to find missing lengths, the ones on the right are used to work out missing angles.

Substitute in the relevant figures to find the missing length or angle.

Exact trigonometric values: