The sine rule

The trigonometric ratios sine, cosine and tangent are used to calculate angles and sides in right angled triangles. We are now going to extend trigonometry beyond right angled triangles and use it to solve problems involving any triangle.

\[\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}\]

We can use the sine rule when we're given the sizes of:

  • two sides and one angle (which is opposite to one of these sides)
  • one side and any two angles


Find the size of angle R.

Diagram of triangle with 75° angle and values 9cm and 4cm

\[\frac{p}{{\sin P}} = \frac{r}{{\sin R}}\]

Substitute the information from the diagram

\[\frac{9}{{\sin (75^\circ )}} = \frac{4}{{\sin R}}\]

Use 'change side, change operation'.

\[9\sin R = 4\sin (75^\circ )\]

\[\sin R = \frac{{4\sin (75^\circ )}}{9}\]


Moving sin to the other side becomes sin-1.


\[R = 25.4^\circ (to\,1\,d.p.)\]

Remember to press 'shift' then 'sin' to get 'sin-1' on your calculator.

Now try the example question below.


Find the length of YZ.

Diagram of triangle with 40° and 95° angles and value of 4cm

We know two angles and so the third angle in the triangle must make up the total \(180^\circ\) ie \(x = 45^\circ\).

\[\frac{x}{{\sin X}} = \frac{z}{{\sin Z}}\]

\[\frac{x}{{\sin (45^\circ )}} = \frac{4}{{\sin (40^\circ )}}\]

\[x = \frac{4}{{\sin (40^\circ )}} \times \sin (45^\circ )\]

\[x = \frac{{4\sin (45^\circ )}}{{\sin (40^\circ )}}\]

\[x = 4\sin (45^\circ ) \div \sin (40^\circ )\]

\[x = 4.40025...\]

Therefore \(YZ = 4.4cm\,(to\,1\,d.p.)\)