# Calculating an angle in a right-angled triangle

### Example

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 $$x^\circ$$.

We know the adjacent and hypotenuse.

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

Substituting the values.

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

$\cos(x^\circ ) = 0.769$

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

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

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