trig

how can you confirm the identity

cos^4x = (1/8)(3+ 4cos2x+ cos4x)

and

sin4x = (4sinxcosx)(2cos^2x-1)

asked by Sarah Kim
  1. You use the identity:

    exp(ix) = cos(x) + i sin(x)

    You can then write:

    cos(x) = [exp(ix)+ exp(-ix)]/2

    Take the fourth power:

    cos^4(x) =

    1/16 [exp(4 i x) + 4 exp(2 i x) + 6 +
    4 exp(-2 i x) + exp(-4 i x)] =

    1/8 {3 + [exp(4 i x) + exp(-4 i x)]/2 + 4 [exp(2 i x) + exp(-2 i x)]/2} =

    1/8 [3 + cos(4 x) + 4 cos(2 x)]

    posted by Count Iblis
  2. sin(4x) can be expanded in the usual way by using the doubling formulas:

    sin(4x) = 2 sin(2x) cos(2x) =

    4 sin(x) cos(x)cos(2x)=

    4 sin(x) cos(x) [2 cos^2(x) - 1]

    The general method also works, but it slightly more laborious in this simple case:

    You start with

    exp(ix) = cos(x) + i sin(x)


    and take the fourth power of both sides and take the imaginary part:

    Im[exp(4 i x)] = 4 cos^3(x)sin(x) -
    4 cos(x) sin^3(x)

    Im[exp(4 i x)] = sin(4 x), so we have:

    sin(4 x) =

    4 cos(x)sin(x)[cos^2(x) -sin^2(x)]

    Using that sin^2(x) = 1 - cos^2(x) gives you the desired result.

    posted by Count Iblis

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math - Trig - Double Angles

    Prove: cos4x = 8cos^4x - 8cos^2x + 1 My Attempt: RS: = 4cos^2x (2cos^2x - 1) + 1 = 4 cos^2x (cos2x) + 1 LS: = cos2(2x) = 2cos^2(2x) - 1 = (cos^2(2)) - cos^2(2x)) - 1 ----- Prove: 8cos^4x = cos4x + 4cos2x + 3 My Attempt: RS: =
  2. precalc

    use power reducing identities to prove the identity sin^4x=1/8(3-4cos2x+cos4x) cos^3x=(1/2cosx) (1+cos2x) thanks :)
  3. Trig identity.

    I need help with verifying these trig identities: 1) sin4x = 4sinxcos - 8sin^3 x cos x 2) cos3x = cos^3 x - 3sin^2 x cos x
  4. Calculus

    Prove the identity and give its domain. 4sinxcosx+sin4x=8sinxcos^3x
  5. Math - Solving Trig Equations

    What am I doing wrong? Equation: sin2x = 2cos2x Answers: 90 and 270 .... My Work: 2sin(x)cos(x) = 2cos(2x) sin(x) cos(x) = cos(2x) sin(x) cos(x) = 2cos^2(x) - 1 cos(x) (+/-)\sqrt{1 - cos^2(x)} = 2cos^2(x) - 1 cos^2(x)(1 -
  6. Trig

    cos4x*cos3x + sin4x*sin3x
  7. Trig.

    Prove sin(4x)= (4sinxcosx)(2cos(x)^(2)-1
  8. MATH

    veify that each equation is an identity cos2x+tan2xcos2x=1 sin4x-cos4x/sin2x=1-cot2x
  9. Calculus

    Evaluate the integral e^x sin4x dx. I know the answer is 1/17 e^x sin4x - 4/17 e^x cos4x + C but I don't know how to solve it out.
  10. Math (Trig)

    Rewrite the following expression in terms of tan3x: (sin2x + sin4x)/(cos2x + cos4x)

More Similar Questions