College Algebra & Trigonometry, 2018a

448 CHAPTER 10. TRIGONOMETRIC IDENTITIES AND EQUATIONS
10.2 Double-Angle Identities
In this section we will include several new identities to the collection we established
in the previous section. These new identities are called “Double-Angle
Identities” because they typically deal with relationships between trigonometric
functions of a particular angle and functions of “two times” or double the original
angle.
To establish the validity of these identities we need to use what are known as
the Sum and Difference Identities. These are identities that deal with expressions
such as sin(α + β). First we will establish an expression that is equivalent to
cos(α − β).
Let’s start with the unit circle:
(cos α, sin α)
α − β
(cos β,sin β)
α
β
(1, 0)
If we rotate everything in this picture clockwise so that the point labeled (cos β,sin β)
slides down to the point labeled (1, 0), then the angle of rotation in the diagram
will be α − β and the corresponding point on the edge of the circle will be:
(cos(α − β), sin(α − β)).
The diagram that represents this rotation is on the next page.