## 2.4 Applications of Trigonometry and Pythagoras Theorem

Note: if you cannot remember the trigonometric identities, revise notes for 2.3 Solving Triangles using Trigonometry.

### Pythagoras’ Theorem

- Pythagoras’ theorem governs the relationship between the lengths of the sides of a right-angled triangle:

a^{2}+b^{2}=c^{2}

Where c is the length of the hypotenuse and a and b are the lengths of the other sides (note that it doesn’t matter which side is chosen to be a and which is chosen to be b).

Example

We wish to find the length of the unknown side in the right-angled triangle above. As the unknown side is not the hypotenuse, we will first have to rearrange our formula to make either a or b the subject. In this case, we will choose a:

a^{2}+b^{2} =c^{2}

a^{2} =c^{2}-b^{2}

a =\sqrt{c^{2}-b^{2}}

Now, we substitute in the known values:

a=\sqrt{15^{2}-8^{2}}=12.69