Expert: Sherman D. - 3/2/2008

Question
I have several Questions
        Solve for exact solutions in interval [0,2Pi)
1. Cos(2x) + Cos(4x)= 0
2. Sin(3x) + Cos(2x)= 0
        Using Half-Angls identities find the exact values
3. Tan(195Degrees)
4. Sin(5Pi/12)
5. Cos(Pi/8)
        Prove these Power-Reducing Identities:
6. Sin^2(u)=(1-Cos(2u))/2
7. Cos^2(u)=(1+Cos(2u))/2
        Prove using Power-Reducing Identities:
8. Cos^3(x)=((1/2)Cos(x))(1+Cos(2x)
9. Sin^5(x)=((1/8)Sin(x))(3-4Cos(2x)+Cos(4x))

I would be very grateful if you could work these out and show me how you did them please I hope to hear from you soon and thanks again.

Answer
1.)
cos(2x) + cos(4x) = 0
cos(2x) + cos(2(2x)) = 0
cos(2x) + 2cos(2x)^2 - 1 = 0
2cos(2x)^2 + cos(2x) - 1 = 0
(2cos(2x) - 1)(cos(2x) + 1)

2cos(2x) - 1 = 0
2cos(2x) = 1
cos(2x) = (1/2)
2cos(x)^2 - 1 = (1/2)
2cos(x)^2 = (3/2)
cos(x)^2 = (3/4)
cos(x) = sqrt(3)/2
x = pi/6, 7pi/6 or -5pi/6, 5pi/6, 11pi/6 or -pi/6

cos(2x) + 1 = 0
cos(2x) = -1
2cos(x)^2 - 1 = -1
2cos(x)^2 = 0
cos(x)^2 = 0
cos(x) = 0
x = pi/2, 3pi/2 or -pi/2

ANS : (±pi/2), (±pi/6), and (±5pi/6)

----------------------------------------------------

2.)
Sin(3x) + Cos(2x)= 0
On this one i used www.quickmath.com to find the answer.

------------------------------------------------------

       Using Half-Angls identities find the exact values
3.)
tan(195°) = tan(390/2)
tan(390/2) = sin(390)/(1 + cos(390)) or sin(30)/(1 + cos(30))
tan(195) = (1/2)/(1 + (sqrt(3)/2))
tan(195) = (1/2)/((2 + sqrt(3))/2)
tan(195) = 1/(2 + sqrt(3))
tan(195) = (2 - sqrt(3))/(4 - 3)
tan(195) = 2 - sqrt(3)

------------------------------------------------

4.)
sin(5pi/12) = sin((5pi/6)/2)
sin((5pi/6)/2) = sqrt((1 - cos(5pi/6))/2)
sin(5pi/12) = sqrt((1 - (-sqrt(3)/2)))/2)
sin(5pi/12) = sqrt(((2 + sqrt(3))/2)/2)
sin(5pi/12) = sqrt((2 + sqrt(3))/4)
sin(5pi/12) = sqrt(2 + sqrt(3))/2

------------------------------------------

5.)
Cos(Pi/8) = cos((pi/4)/2)
cos((pi/4)/2) = sqrt((1 + cos(pi/4))/2)
cos(pi/8) = sqrt((1 + (sqrt(2)/2))/2)
cos(pi/8) = sqrt(((2 + sqrt(2))/2)/2)
cos(pi/8) = sqrt((2 + sqrt(2))/4)
cos(pi/8) = sqrt(2 + sqrt(2))/2

----------------------------------------------------

       Prove these Power-Reducing Identities:
6.)
sin(u)^2 = (1 - cos(2u))/2
sin(u)^2 = (1 - (1 - 2sin(u)^2))/2
sin(u)^2 = (1 - 1 + 2sin(u)^2)/2
sin(u)^2 = sin(u)^2

----------------------------------

7.)
cos(u)^2 = (1 + cos(2u))/2
cos(u)^2 = (1 + (2cos(u)^2 - 1))/2
cos(u)^2 = (2cos(u)^2)/2
cos(u)^2 = cos(u)^2

-----------------------------------------

       Prove using Power-Reducing Identities:
8.)
cos(x)^3 = ((1/2)cos(x))(1 + cos(2x))
cos(x)^3 = (1/2)cos(x)(1 + (2cos(x)^2 - 1))
cos(x)^3 = (1/2)cos(x)(2cos(x)^2)
cos(x)^3 = cos(x)^3


-------------------------------------------------------------

9.)
sin(x)^5 = ((1/8)sin(x))(3 - 4cos(2x) + cos(4x))
sin(x)^5 = (1/8)sin(x)(3 - 4(1 - 2sin(x)^2) + (1 - 2sin(2x)^2))
sin(x)^5 = (1/8)sin(x)(3 - 4 + 8sin(x)^2 + 1 - 2(2sin(x)cos(x))^2)
sin(x)^5 = (1/8)sin(x)(8sin(x)^2 - 2(4sin(x)^2cos(x)^2))
sin(x)^5 = (1/8)sin(x)(8sin(x)^2 - 8sin(x)^2(1 - sin(x)^2))
sin(x)^5 = (1/8)sin(x)(8sin(x)^2 - 8sin(x)^2 + 8sin(x)^4)
sin(x)^5 = (1/8)sin(x)(8sin(x)^4)
sin(x)^5 = sin(x)^5

info found at http://math2.org/math/trig/identities.htm