Compound Angle Formulæ

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We want to find a way of handling expressions such as

You do not need to know how the formulæ are derived, but an explanation is given below, for your interest. To go straight to the rules and examples, click here

The formula for

Consider the diagram below.

The side AB has a length of 1. This leads to the equation

so, we need to find the lengths BF and FE.

Let's first look at the triangle ABC,( Note that angle BCA is a right angle.)

This leads us to

and

With a little thought, you can see that the angle FBC must equal

(Clue: FCA is an alternate (Z) angle.)

Let's update the diagram

Remember now, that

represents a length. We can use these two lengths to calculate BF and FE.

First consider the triangle BCF

Now consider the triangle ACD

which means that

Therefore we can say that

This is a standard identity and may be quoted

On the next page, a similar identity is derived for the cosine of a compound angle

Go to Derivation of cos

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