5.4 Multiple-Angle Identities
Name: ________________
Objectives: Students will be able to apply the double-angle identities,
power-reducing identities, and half-angle identities.
DOUBLE ANGLE IDENTITIES
sin2θ = 2sinθcosθ
cos2θ = cos2θ - sin2θ
cos2θ = 2cos2θ - 1
cos2θ = 1 - 2sin2θ
tan2θ = 2tanθ
1 - tan2θ
Dec 3­2:50 PM
HALF-ANGLE IDENTITIES
sin(θ/2) = ± 1 - cosθ
2
cos(θ/2) = ± 1 + cosθ
2
tan(θ/2) = ± 1 - cosθ = 1 - cosθ = sinθ
1 + cosθ
sinθ
1 + cosθ
Dec 3­3:04 PM
1
Power-Reducing Identities
sin2x = 1 - cos2x
2
cos2x = 1 + cos2x
2
tan2x = 1 - cos2x
1 + cos2x
Dec 6­1:18 PM
Examples Use the appropriate sum or difference identity to
prove the double-angle identity.
1.) cos2u = 2cos2u - 1
2.) tan2u = 2tanu
1 - tan2u
Dec 6­1:08 PM
2
Examples Find all solutions to the equation in the interval [0,2π].
3.) cos2x = cosx
4.) cos2x + cosx = cos2x
Dec 6­1:11 PM
5.) Write the expression as involving sinθ and cosθ.
sin3θ + cos2θ
Dec 6­1:12 PM
3
Prove the identities.
6.) cos6x = 2cos23x - 1
7.) sin3x = sinx(3 - 4sin2x)
Dec 6­1:14 PM
8.) Solve algebraically for exact solutions in the interval
[0,2π). Use your graphing calculator to support your algebraic
work.
sin3x + cos2x = 0
Dec 6­1:15 PM
4
Use half-angle identities to find an exact value without a
calculator.
10.) sin(5π/12)
11.) tan195o
Dec 6­1:16 PM
Use power-reducing identities to prove the identity.
12.) cos3x = (1/2)cosx(1 + cos2x)
13.) sin32x = (1/2)sin2x(1 - cos4x)
Dec 6­1:20 PM
5
Assignment: 1 - 41 odd
Dec 6­1:22 PM
6