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Math Help - tanxsinx = 3sinx - secx

  1. #1
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    tanxsinx = 3sinx - secx

    Solve for x:

    equal to or less than 360
    equal to or more than 0

    tanxsinx = 3sinx - secx

    A problem I have been struggling with for ages.
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  2. #2
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    Quote Originally Posted by differentiate View Post
    Solve for x:

    equal to or less than 360
    equal to or more than 0

    tanxsinx = 3sinx - secx

    A problem I have been struggling with for ages.
    not very nice, I agree.

     <br />
\tan{x}\sin{x} = 3\sin{x} - \sec{x}<br />

    \frac{\sin{x}}{\cos{x}} \cdot \sin{x} = 3\sin{x} - \frac{1}{\cos{x}}

    \frac{\sin^2{x}}{\cos{x}} = \frac{3\sin{x}\cos{x} - 1}{\cos{x}}

     <br />
\sin^2{x} = 3\sin{x}\cos{x} - 1<br />

    \sin^2{x} = 3\sin{x}\cos{x} - (\sin^2{x} + \cos^2{x})

     <br />
2\sin^2{x} - 3\sin{x}\cos{x} + \cos^2{x} = 0<br />

     <br />
(2\sin{x} - \cos{x})(\sin{x} - \cos{x}) = 0<br />

    2\sin{x} = \cos{x} or \sin{x} = \cos{x}

    for \sin{x} = \cos{x} ...

    x = 45^{\circ} , x = 225^{\circ}

    for 2\sin{x} = \cos{x}

    4\sin^2{x} = cos^2{x}

     <br />
4\sin^2{x} = 1 - \sin^2{x}<br />

     <br />
5\sin^2{x} - 1 = 0<br />

     <br />
\sin^2{x} = \frac{1}{5}<br />

     <br />
\sin{x} = \pm \sqrt{\frac{1}{5}}<br />

    the solutions to this part were found using a calculator and looking at a graph ...

    x = \arcsin\left(\sqrt{\frac{1}{5}}\right)

    and

     <br />
x = \arcsin\left(\sqrt{\frac{1}{5}}\right) + 180^{\circ}<br />
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