Double Angles and Half Angles

In my article Starting with the Cosine Difference Formula I promised that this was just the beginning. The next thing to do with the sum and difference formulas is to get the double and half angle formulas from them. Out of the sum formulas we can get double angle formulas and then half angle formulas and other useful results.

Double Angle Identities

Multiplication is repeated addition, so and This cosine identity is especially interesting because you can write it in two other different forms. Because it is only in terms of squares of sine and cosine, it is easy to write it all in terms of either one of them. sin²=1-cos² and cos²=1-sin², so we get and We can also get an identity for tangent from the sum identity.

Here are the double angle identities all together.

Half Angle Identities

Now here's the really interesting thing about the forms that the cosine double angle formula can be put in. Because they can be written all in terms of one function, they can be used to get half angle identities. You see if you solve for sin(theta) or cos(theta) in terms of cos(2theta) that would really be a half angle formula in disguise since you are using the knowing of a trig function at one angle to get one at an angle that is half as large. Also when we do this along the way we also get a couple of other identities that are particularly useful in calculus.

So, here we go.

These two identities will be useful on their own, particularly if you take calculus, so they are worth remembering. To get the half angle identities we have to go on and take square root, which is kind of annoying because we get a plus or minus. Then to make these truly look like half angle formulas, we have to make a substitution and let phi=theta/2. Doing that we get our identities. The plus or minus is a little bit annoying, but when you are working with actual angle values you will know which way it is by the quadrant it is in.

Out of the sine and cosine identities you can also get a tangent identity.

By rationalizing the denominator we can get this into a much nicer form. I'm going to cheat a bit with this at first and ignore all of the plus or minuses and anything else like absolute value that has to do with sign and I will fix it up later. We've been doing some illegal stuff, but it turns out it is right anyway and the easiest way is to simply check the signs quadrant by quadrant. Notice that the denominator has to be positive, so all we have to do is check to see if tan(phi/2) and sin(phi) always have the same signs.

• Case 1: phi/2 is in the 1st quadrant. phi must be in the 1st or the 2nd quadrant. tangent is + in the first quadrant. sine is + in the 1st and 2nd quadrants.
• Case 2: phi/2 is in the 2nd quadrant. Then phi must be in the 3rd or 4th quadrants. tangent is - in the 2nd quadrant. sine is - in the 3rd and 4th quadrants.
• Case 3: phi/2 is in the 3rd quadrant. Then phi must be in the 1st or 2nd quadrants, next rotation around. tangent is + in the 3rd quadrant, because both sine and cosine are minus there. sine is + in the 1st and 2nd quadrants.
• Case 4: phi/2 is in the 4th quadrant. Then phi must be in the 3rd or 4th quadrants, next rotation around. tangent is - in the 4th quadrant. sine is + in the 1st and 2nd quadrants.

If you rationalize the numerator instead of the denominator you get a different form for it. I'll let you work out the details. Believe it or not they really are equal. You can prove that directly if you want by multiplying the first one top and bottom by 1-cos(phi)

Here are all of the  half angle identities put together for you.

Extra for Experts

An interesting exercise for getting used to using these identities is to find identities for sines and cosines of larger multiples of theta like 3theta and 4theta. With enough patience you could work out this sort of thing for any multiple, but for large multiples it would get pretty messy. Here's how you would do it for cos(3theta).