Trigonometry helps solve problems involving right-angled triangles using the sine, cosine or tangent ratios. SOH CAH TOA is used to help remember the formulae.

Part of

Calculate \(x^\circ\) to one decimal place.

First label the sides.

As the numbers are known on the opposite and the hypotenuse, then we look for the ratio which uses both these sides (SOH CAH TOA). That ratio is the sine ratio.

\[\sin (x^\circ ) = \frac{{opposite}}{{hypotenuse}}\]

Take care to make sure you put the numbers in the correct position!

\[\sin (x^\circ ) = \frac{8}{{10}}\]

\[\sin (x^\circ ) = 0.8\]

Rearrange using 'change side, change operation'. When you move the 'sin' over to the other side of the equals sign, we do the opposite which is sin^{-1} (inverse sin).

\[x^\circ= sin ^{-1 (0.8)}\]

On your calculator, type 'shift' then 'sin' to get sin^{-1}

\[x^\circ = 53.130...\]

Answer is then rounded to the appropriate figure of accuracy.

\(x^\circ = 53.1^\circ\) (one decimal place).

Try this example.

- Question
Calculate \(x^\circ\).

Give your answer to one decimal place.

We know the adjacent and hypotenuse.

\[\cos (x^\circ ) = \frac{{adj}}{{hyp}}\]

Substituting the values.

\[\cos (x^\circ ) = \frac{{20}}{{26}}\]

\[\cos(x^\circ ) = 0.769\]

\[x^\circ = \cos ^{-1}(0.769)\]

\[x^\circ = 39.7^\circ (to\,1\,d.p.)\]