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

When answering a trigonometry problem:

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

Calculate \(y\).

Give your answer correct to one decimal place.

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.

- Question
Calculate y.

Give your answer to three decimal places.

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

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

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

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

\[y = 15.37727...\]

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