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

cos2θ = 2 cos2θ - 1



Example: If

cos2x = 1 - 2 sin2 x

Substitute

Simplify

Simplify

Simplify

Example: Find all the solutions of 2 cosθ + sin2θ = 0     where

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°

sin θ = -1 when

Therefore the solution set is 90°



Related Links:
Math
Trigonometry
Half Angle formulas
Even and Odd Trig Functions
Sum and Difference of Angles Identities


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