Trigonometry examples

When answering a trigonometry problem:

  1. label the sides on the triangle
  2. decide which ratio to use (SOH CAH TOA)
  3. substitute the correct information into the ratio
  4. rearrange to find '\(x\)'
  5. solve using your calculator making sure your calculator is set to 'degrees' mode


Calculate \(y\).

Give your answer correct to one decimal place.

Right-angled isosceles triangle with a 42° base angle and a base of 12cm


We know the hypotenuse and are trying to find the value of \(y\), which is the adjacent.

From SOH CAH TOA, we see that we need to use the cosine ration.

\[\cos (x^\circ ) = \frac{{adjacent}}{{hypotenuse}}\]

In this case we have \(\cos (42^\circ ) = \frac{y}{{12}}\)

Rearrange using 'change side, change operation'. We need to move the '12' over to the other side of the equals sign so that we have 'y' on its own. The '12' is dividing on the right hand side, so when it moves to the other side it does the opposite, therefore it will multiply.

\[12 \times \cos (42^\circ ) = y\]

\[y = 8.917...\]

\[y = 8.9cm\,(to\,1\,d.p.)\]

Remember to show all working, especially when you use a calculator.


Calculate y.

Give your answer to three decimal places.

Right-angled scalene triangle with a 64° angle and a height of 7.5cm

We know the adjacent and we are trying to find the opposite.

\[\tan(x^\circ ) = \frac{{opp}}{{adj}}\]

Substituting the values \(\tan (64^\circ ) = \frac{y}{{7.5}}\)

\[7.5 \times \tan (64^\circ ) = y\]

\[y = 15.37727...\]

\[y = 15.377cm(to\,3\,d.p.)\]