Create new identities...using the ones we already have.
Use the angle sum identity to express sin(2A).
Use the angle sum identity to express cos(2A).
Section 5.3: Double Angle Identites
sin 2A = 2 sin A cos A
tan 2A =
2tan A
1 ­ tan2A
2
2
cos 2A = cos A ­ sin A
= 2cos2A ­ 1
= 1 ­ 2 sin2A
Example: Let's practice using this identity to find sin90o
Find the exact value for cos 600o
Example: Use a double angle formula to find the exact value of tan 2A
if cos A = ­3/4 and A is in quadrant 3.
2
2
A
cos 2A = cos A ­ sin
= 2cos2A ­ 1
= 1 ­ 2 sin2A
Example: If tanθ = 2/3 and θ is in quadrant 3, then find cos2θ.
Simplify each of the following by recognizing the form of an identity.
2
2
cos 2A = cos A ­ sin A
= 2cos2A ­ 1
2sin10ocos10o
= 1 ­ 2 sin2A
sin 2A = 2 sin A cos A
cos2 π/8 ­ sin2 π/8