Calculating an angle in a right-angled triangle

Example

Calculate x^\circ to one decimal place.

Right-angled scalene triangle with an x° angle, opposite of 8 and a hypotenuse of 10

Answer

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.

Right-angled scalene triangle with an x° angle, adjacent of 20 and a hypotenuse of 26

We know the adjacent and hypotenuse.

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

Substituting the values.

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

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

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

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

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