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March 27, 2017

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Solve for x:

sec(2x)csc(2x) = 2csc(x)

for 0 < x < 2pi

[the first "<" sign is less than or equal to]

Thank you!

  • Math - ,

    sec(2x)csc(2x) = 2csc(x)
    1/((cos 2x)(sin 2x)) = 2/sinx
    cross-multiply
    2(cos 2x)(sin 2x) = sinx
    2(cos 2x)(2sinxcosx) = sinx
    divide by sinx
    2(cos 2x)(2cosx) = 1
    4cosx(2cos^2 x - 1) = 1
    8cos^3 x - 4cosx - 1 = 0

    let's let cosx = a
    so we are solving
    8a^3 - 4a - 1 = 0

    after a few tries I go a = -1/2 to work
    giving me
    (2a-1)(4a^2 - 2a - 1) = 0

    a = -.5 or a = .809 or -.309

    so cosx = -.5 or cosx = .809 or -.309

    I will do one of these:
    cosx = -.5, the reference angle is pi/3 or 60º
    but the cosine is negative in quadrants II and III
    so x = pi - pi/3 = 2pi/3 or 180-60 = 120º
    or x = pi + pi/3 = 4pi/3 or 180+60 = 240º

    You should have 6 different answers

  • Math - ,

    Left hand side -

    sec(2x)csc(2x)= 1/ cos(2x) 1/sin(2x)

    = 1/(Cos2x)* 1/(2 Sinx Cosx)

    Right Hand side = 1/2Sinx

    Equating LHS & RHS

    1/{Cos2x}* 1/(2 Sinx Cosx)
    = 1/2Sinx

    or
    1/{Cos2x}* 1/cos x = 1

    or {Cos2x}* Cosx = 1

    this actually only holds true for x = 0 and x = 2pi

  • Vipster - Math - ,

    Vipster, if x = 0, the original equation has undefined calculations, csc 0 is undefined.

    you have an error by saying
    2csc(x) = 1/2Sinx

    2csc(x) = 2/sinx

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