3. Double-Angle Formulas
by M. Bourne
The double-angle formulas can be quite useful when we need to simplify complicated trigonometric expressions later.
With these formulas, it is better to remember where they come from, rather than trying to remember the actual formulas. In this way, you will understand it better and have less to clutter your memory with.
Sine of a Double Angle
Recall from the last section, the sine of the sum of two angles:
sin(α + β) = sin α cos β + cos α sin β
We will use this to obtain the sine of a double angle.
If we take the left hand side:
sin(α + β)
and replace β with α, we get:
sin(α + β) = sin(α + α) = sin 2α
Consider the RHS:
sin α cos β + cos α sin β
If we replace β with α, we obtain:
sin α cos α + cos α sin α = 2 sin α cos α
Putting our results for the LHS and RHS together, we obtain the important result:
sin 2α = 2 sin α cos α
This result is called the sine of a double angle. It is useful for simplifying expressions later.
Cosine of a Double Angle
Using a similar process, we can start with the cosine of the sum of two angles
cos(α + β) = cos α cos β − sin α sin β,
and obtain the cosine of a double angle formula:
cos 2α = cos2 α − sin2 α
By using the result sin2 α + cos2 α = 1, (which we found in Trigonometric Identities) we can write the RHS of the above formula as:
cos2 α − sin2 α
= (1 - sin2 α) − sin2 α
= 1 - 2sin2 α
Likewise, we can substitute (1 − cos 2 α) for sin2 α into our RHS and obtain:
cos2 α − sin2 α
= cos2 α − (1 − cos2 α)
= 2cos2 α − 1
Summary - Cosine of a Double Angle
The following have equivalent value, and we can use whichever one we like, depending on the situation:
cos 2α = cos2 α − sin2 α
cos 2α = 1 - 2sin2 α
cos 2α = 2cos2 α − 1
Example 1
Find cos 60° by using the functions of 30°.
Example 2
Find cos 2x if sin x = -12/13 (in Quadrant III).
Exercises
1. Without finding x, find sin 2x if cos x = 4/5 (in Quadrant I).
2. Prove that
3. Prove that
2 csc 2x tan x = sec2x
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