The sine rule can be used to find an angle from 3 sides and an angle, or a side from 3 angles and a side. The cosine rule can find a side from 2 sides and the included angle, or an angle from 3 sides.

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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.

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

Substitute the information from the diagram

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

Use 'change side, change operation'.

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

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

\[SinR=0.429\]

Moving sin to the other side becomes sin^{-1}.

\[R=sin^{-1}\,0.429\]

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

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

Now try the example question below.

- Question
Find the length of YZ.

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