The Sine

The Sine & Cosine Rules

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We have seen how we can use sin, cos and tan, to find missing sides and angles in right-angled triangles.

We will now look at the two rules that let you find sides and angles in non right-angled triangles.

You do not need to know the derivations of these rules, and they will probably be given in your formula book, but you should learn them anyway, as this will make them easier to apply

To skip straight to the rules and examples, press here

The Sine Rule

Consider the triangle below

The large triangle,( the blue triangle,) can be split into two right-angled triangles, with a height h.

We can now use trig on the two right-angled triangles

Equating these two expressions gives

which will work for any triangle

If you are looking for an angle, you can of course turn the formula upside down

The Cosine Rule

Consider the triangle below

As before, we have split the triangle into two right-angled triangles.

We have now said that the bottom side, c, has been split into sections, of length x and c-x.

Using Pythagoras on the two triangles gives

Combining these gives

However, we can use trigonometry to get rid of the x.

This can be substituted into the previous equation.

Which can be re-arranged to the cos rule.

If you want to find an angle, the formula can be re-arranged to the form

The rules are summarized and demonstrated on the next page.

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