Solve the equation on [0,2π)
2cos2θ-1 + cosθ + 1 = 0
2cos2θ + cosθ = 0
cosθ(2cosθ + 1) = 0
cosθ = 0 2cosθ+1=0
cosθ=-1/2
π/2, 3π/2
2π/3, 4π/3
1
Solve the equation on [0,360o)
sinxcos(π/4) + cosxsin(π/4) + sinxcos(π/4) - cosx sin(π/4)=-1
2sinxcos(π/4)=-1
2sinx(√2/2)=-1
√2sinx=-1
sinx= -√2/2
225o, 315o
2
Solve:
2cos2x = 1
cos2x = 1/2
cosx = ±√2/2
π/4 + πn
3π/4 +πn
n∈ZZ
3
Solve:
Sin2x ­ sinx ­2=0
(sinx­2)(sinx + 1) = 0
sinx­2=0 sinx +1=0
sinx=2 sinx=­1
3π/2 +2πn n∈ZZ
None
4
Verify:
sec2xsin2x pythagorean identities
(1/cos2x)sin2x reciprocal identities
sin2x/cos2x multiply fractions
quotient identities
tan2x
5
Verify:
sin(3π-x) = sinx
sin3πcosx - cos3πsinx sum and difference formula
0cosx - (-1)sinx
evaluate trig functions
0+sinx
sinx
simplify
6
cos2θ = 1-sin2θ
cos2θ = 1-(2/3)2
cos2θ=1-4/9
cos2θ = 5/9
cosθ = -√5/3
7
Use the triangle to find cos(2θ)
tan(2θ)
2
5
θ
cos(2θ)= 1-2sin2θ = 1-2(2/5)2 = 1-2(4/25) = 1-8/25 = 17/25
tan(2θ) = sin(2θ)/cos(2θ)
sin(2θ) = 2sinθcosθ = 2(2/5)(√21/5)= 4√21/25
tan(2θ) = 4√21/25÷17/25= 4√21/17
8