cos 2A
How to find cos 2A by using its formula (double-angle formula): formula, 2 examples, and their solutions.
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
Examplecos θ = 1/4, cos 2θ = ?
Solution
Solution (Detail)
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.
Examplesin θ = -2/3, cos 2θ = ?
Solution
Solution (Detail)
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.