Double Angle Identities

Double angle identities involve reducing the trigonometric function of a doubled angle (2x, for example) and expressing it in terms of its single component (in this example, x). To do this we must first know the sum formula identities:

Since

,

we can use the sum formula identities to derive a double angle identity.

Here is the derivation for the double angle identity for sine:

Here is the derivation for the double angle identity for tangent:

And finally, the derivation for the double angle identity for cosine. This one is the trickiest because it has three possible derivations that can be used in different situations.

Here is the first of the three derivations:

From here, we can derive two more equivalent identities using the Pythagorean Identities to derive two more:

Version 1 (cosine only):

Version 2 (sine only):

Here is a picture of all the double angle identities together:

Important things to remember:

• trig(2x)=/= 2trig(x)
• There are three derivations for cos(2x)

In loving memory of John D. Enison