cos 2A

How to find cos 2A by using its formula (double-angle formula): formula, 2 examples, and their solutions.

Formula

Formula

cos 2A
= 2 cos2 A - 1
= cos2 A - sin2 A
= 1 - 2 sin2 A

These are the double-angle formulas of cosine.

To prove these formulas,
put A and A into cos (A + B) formula:
cos (A + A)
= cos A cos A - sin A sin A
= cos2 A - sin2 A.

The other formulas can be found
by using sin2 A + cos2 A = 1.

Pythagorean Identity

Example 1

Example

Solution

It says to find cos 2θ.

And cos θ is given:
cos θ = 1/4.

So cos 2θ = 2⋅(1/4)2 - 1.

(1/4)2 = 1/16

2⋅[1/16] = 1/8

Change -1 to -8/8.

1/8 - 8/8 = -7/8

So cos 2θ = -7/8.

Example 2

Example

Solution

It says to find cos 2θ.

And sin θ is given:
sin θ = -2/3.

So cos 2θ = 1 - 2⋅(-2/3)2.

(-2/3)2 = 4/9

Change 1 to 9/9.

-2⋅[4/9] = -8/9

9/9 - 8/9 = 1/9

So cos 2θ = 1/9.