Trig

Find the general solution for the equation 4tanx+cotx=5?????? and the steps please? THANYOO!

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  1. For these I generally like to change everything to sines and cosines

    4tanx + cotx = 5
    4sinx/cosx + cosx/sinx = 5
    (4sin^2x + cos^2x)/sinxcosx = 5
    4sin^2x + cos^2x = 5sinxcosx
    4sin^2x - 5sinxcosx + cos^2x = 0
    (4sinx - cosx)(sinx - cosx) = 0

    4sinx = cosx
    sinx/cosx = 1/4
    tanx = 1/4
    x = 14.0° or 194.0°

    or

    sinx = cosx
    sinx/cosx = 1
    tanx=1
    x = 45° or 225°

    the period of tanØ is 180°

    general solutions:
    45° + 180k°
    14° + 180k°

    notice that 45+180 = 225 , the other answer
    and 14+180 = 194 , also the other answer, so we don't have to generalize those, they get hit anyway.

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