We proved the sine addition formula; now we're going to prove the cosine addition formula.
Before we do that we will talk about subtraction formulas.
You do not need to learn or remember special subtraction formulas or 'angle difference' formulas for sine and cosine. You can work them out 'instantly' from the addition formulas for sine and cosine, using and .
Let's put in place of in the two addition formulas:
First the sine addition formula:
Now for the cosine addition formula:
To be really pedantic, instead of just putting in place of in the equations we could/should have done something more like saying: "Let's choose " and then we'd have got formulas for and , which would come to the same thing. We took a perfectly OK shortcut to re-use the letter.
In maths we often think along these lines: "The original formula is true for all and all ; I can put any expression in I like in place of or ". We could for example substitute into the addition formula for cosines (reusing our letters) and write:
and it is true. It is true for all and - it is just not a particularly useful substitution to make.
Combining all four formulas:
If we really want to we can write the four addition and 'angle difference' formulas in a more condensed notation like so:
If you like this style, use them. We'd recommend instead just learning the addition formulas and deriving the difference formulas from them when you need them.
We can in fact derive one from the other without using a diagram at all.
Worked Example: Cosine Addition Formula from Sine Addition Formula
We use and and (substituting in several places):
Now we use and
This is true for all and so if we put and we get:
Now it is your turn to practice deriving new formulas from old ones:
Exercise: Sine Addition Formula from Cosine Addition Formula
A somewhat harder exercise:
Exercise: Tangent Addition Formula
and the addition formulae for sin and cos, show that
And now it is your turn to do the geometric proof of addition formulas.
Exercise: Using a different Diagram for the proof
You might want to skip this exercise and come back to it later after you have used the cosine addition formula for a bit. It is a good exercise for getting to the stage where you are confident you can write a geometric proof of the formulas yourself.
Start from the diagram below:
Add labels to it, and write out a proof of
Sine addition formula
Cosine addition formula
based on the diagram and the letters you have chosen. Make sure you explain by chasing angles why the two angles labelled are the same. The labels given to the edge lengths are to help you. Your proof must spell out why those labels are correct, using the trig relations.
Compare the diagram with the one in the proof above. Just how different are they really?