The Double-Angle formulas express the cosine and sine of twice an angle in terms of the cosine and sine of the original angle. We are going to derive them from the addition formulas for sine and cosine. The formulae are:
It is well worth practising the derivation so that you can do it quickly and easily. Then you will not need to remember the formulae, since you can get them quickly from the addition formulas for sine and cosine. It is also good to practice the derivation because being more fluent with the algebra will make you better at other algebra used with trigonometry.
Exercise: Check these Make Sense
Did we make a typo in these formulae? Check they at least make sense.
We know that and . Try those values out. Do the formulae work?
and . Are the formulae consistent with that?
Make up your own additional 'spot check' to check the formulae
Example: Half Angle formula for Cosine
If we put we immediately get
Check it. Do you agree? Or rearranging:
So, if we know the cosine of (we do, it is zero), we can compute the cosine of and and and so on.
We'll prove the double angle formulae from the addition formulae. Recall that:
Putting in the above formula yields:
Compare this with the "Pythagorean Theorem" expressed in terms of sine and cosine. Notice the double angle formula above has a minus not a plus, otherwise it would be saying , which would mean cos was 1 for all values of t, which we know is not true.
Using the above procedures twice, and the Pythagorean theorem where appropriate, we find
By repeating the procedure, we can find formulae for and for any integer n. The formulas do however get rather long.
It is not really worthwhile remembering these formulae. They are not used often, and can either be looked up in tables of formulae or calculated when you need them. It is quite good for practising algebra to derive them yourself, so....
Exercise: Treble angle formulas for sine and cosine
Like the formulas for and , these formulae aren't often useful, but again it is good to be able to work them out yourself.
Exercise: Multiple angle formulae for tan
Derive the formulas for and yourself.
If you don't get 'the right answer,' don't panic. It takes time and practice to become fluent at algebraic manipulations. Here, where you know the right answer, you can work through your steps and try and find for yourself where you went wrong. Very often it is a sign error somewhere, adding a value instead of subtracting, and then everything from that point on is wrong. You can use the trick of putting in actual values for (and calculating with a calculator) to check for the place where the first error creeps in.