Double Angle Identities

Double angle identities can be used to solve certain integration problems where a double formula may make things much simpler to solve. Notice that there are several listings for the double angle for cosine. That's because you can substitute for either of the squared terms using the basic trigonometric identity sin²θ + cos²θ = 1.

cos2 θ = cos²θ - sin²θ

sin2 θ = 2sin θ · cosθ cos2θ = 1 - 2 sin²θ $Double angle identities 1$

cos2θ = 2 cos²θ - 1

Example: If $Double angle identities 2$

cos2x = 1 - 2 sin² x

Substitute $Double angle identities 3$

Simplify $Double angle identities 4$

Simplify $Double angle identities 5$

Simplify $Double angle identities 6$

Example: Find all the solutions of 2 cosθ + sin2θ = 0 where $Double angle identities 7$

1) Replace sin2θ with 2sin θ · cos θ

2cosθ + 2sin θ · cos θ = 0

2) Factor out the common factor of 2 cosθ

2cos θ (1 + sinθ) = 0

3) Use the zero product rule

2 cos θ = 01 + sinθ = 0

cosθ = 0 sin θ = -1

4) Determine when cos θ = 0 and sin θ = -1

cos θ = 0 when θ is 90° $Double angle identities 8$

sin θ = -1 when $Double angle identities 9$

Therefore the solution set is 90° $Double angle identities 8$

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