Calculate Side Lengths on a Triangle Using Trigonometry

In a right-angled triangle, the sides and the angles are connected by three trigonometric ratios.

 

 

The three trig ratios are:

 

 

The best way to use these is in a formula triangle:

 

For example with sin:

 

 

The triangle above shows the formula triangle for using sin. 

To use it, we simply cover up the part of the triangle that we want to find out and then rearrange the equation.

 

For example, if we know the sin and the opposite side, we cover up H (the hypotenuse) and we can see that we need to do: O ÷ S = H see below

 

 

If we know the sin and the hypotenuse, we cover up O and we can do this equation: S x H = O see below:

 

 

Clearly, if we know O and A but don't know S, we'd cover up S and do O ÷ H = S

 

We can remember the triangle by using the letters SOH to spell a sort of word!

 

We can do the same for the other two formulae too:

 

Here is the triangle for cos - we can remember this one by the word CAH

 

 

And here's the one for tan - we can remember this one by the word TOA

 

 

We can stick the words together to make a pretty silly-looking word:

 

SOHCAHTOA 

 

Example 1

Using trigonometry, calculate the side length x to 3 sig. figs.

 

 

Answer

Label the sides adjacent, opposite and hypotenuse in relation to the given angle 34º.

The given sides are the opposite (x) and the hypotenuse (8). We don't need to know the adjacent side.

 

So, we know the hypotenuse and the angle, and we need to find out the opposite side. So which formula do we need that has S, O and H in it?

It's the SOH one with SIN!

 

 

Which letter do we need to cover up in the triangle? The side we want to find out is O, so cover up O and we do H x Sin

 

x = 8 sin34º

x = 8 x  0.559192903....

x = 4.47 (3 s.f.)

 

 

Example 2

 

Using trigonometry, calculate the side length x to 3 sig. figs.

 

 

Answer

Label the sides adjacent, opposite and hypotenuse in relation to the given angle 54º.

The given sides are the hypotenuse (x) and the adjacent (7). We don't need to know the opposite side.

 

So we know the angle and the adjacent side and we need to know the hypotenuse. Which formula do we need? It's the CAH one with COS.

 

 

Look at the CAH triangle above. We want to find the hypotenuse, so cover up H and we do A ÷ cos

 

x = 7 ÷ cos54º

x = 7 ÷  0.587785252....

x = 11.9 (3 s.f.)

 

It looks very complicated but really it's very simple once you've chosen the correct formula triangle to use!

 

If you need to remind yourself of the formula triangles, just click on the red button at any point during the activity and this introduction will magically appear!!

 

Let's get started!