Mathematicians have to calculate the unknown side lengths or angles in a right triangle all the time. To do this, they apply the trigonometric functions.

For example, if you know the value of angle θ, you can find the two sides of a right-angled triangle.

###### Trigonometric Function Example

Suppose you have a right triangle with:

- Angle θ = 30 degrees
- Adjacent side =
*a* = 5

You want to find the length of the opposite side *b*.

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:

tan(θ) = opposite/adjacent

You know that:

- θ = 30
- Adjacent side
*a* = 5

So, using the tangent function looks like this:

You know from trigonometric tables or by using a calculator that:

So:

Now, to find *b*:

The length of the opposite side *b* is approximately 2.885 units.