Trigonometry, which originally meant the study of triangles, is centred around three important ratios, which give us a relationship between the side lengths and the angles in a right-angled triangle. These ratios must be memorised as they underlie all of the work on trigonometry that we will do.

Using this relationship between angles and sides in a right-angled triangle, we can solve problems where we know some information about a right-angled triangle and want to find out other information. Let’s start by using it to find the length of some missing sides.

Example

Find the lengths of all of the sides labelled x below:

Exercise

Let’s complete exercise 2 on pages 210 to 212, where in each question we must find a missing side on a right-angled triangle:

The answers are below:

Extension Exercise

Let’s complete exercise 3 on pages 212 to 213, again identifying missing lengths:

The answers are below:

Now let’s try using these same ratios to find missing angles. To do this we will be using the arccos function on our calculators, which is labelled as cos^-1 and because it is above the cos function, we get it by pressing shift, cos.

Example

Exercise

Let’s complete exercise 4 on pages 214 to 216 to practice calculating a missing angle using trigonometry:

The answers are below:

Further Mixed Questions

Answers