cos 2A (Double-Angle Formula)

cos 2A (Double-Angle Formula)

How to solve cos 2A (Double-angle formula) problems: formula, proof, examples, and their solutions.

Formula

cos 2A = cos^2 A -  sin^2 A = 2 cos^2 A - 1 = 1 - 2 sin^2 A

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

In most cases,
the latter two formulas are more frequently used,
because you only need one variable to find cos 2A.

(To use the first formula,
you need two variables: both cos A and sin A.)

Proof

cos 2A (Double-Angle Formula): Proof of the Formula

cos 2A = cos (A + A)
= cos A cos A - sin A sin A

cos (A + B)

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

Pythagorean identities

So cos 2A = 2 cos2 A - 1.

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

Pythagorean identities

So cos 2A = 1 - 2 sin2 A.

Example 1

If cos theta = 1/4, find the value of cos 2*theta.

Cosine is given:
cos θ = 1/4.

So use cos 2θ = 2 cos2 θ - 1:

cos 2θ = 2⋅(1/4)2 - 1.

Example 2

If sin theta = -2/3, find the value of cos 2*theta.

Sine is given:
sin θ = -2/3.

So use cos 2θ = 1 - 2 sin2 θ:

cos 2θ = 1 - 2⋅(-2/3)2.