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Math Help - sovle for x, use identity

  1. #1
    apm
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    sovle for x, use identity

    i've figured out part of the problem but i dont' understand what part of it means and i don't know how to do it without a calculator

    3 sin^2 x=cos^2 x 0<or equal to x <2pi - what does this inequality mean?

    i know that sin^2 x + cos^2 x = 1 so i did this

    3 sin^2 x = 1 - sin^2 x
    but i don't know what how to solve for x from here and i can't use a calc

    thanks!
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  2. #2
    Senior Member Twig's Avatar
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    hi

    Hi!

     \mbox{Let} \ sin(x) \ = t

    Can you solve it from here?
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  3. #3
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    Here's a slightly different method.

    Simply divide by \cos^2{x} on both sides and divide by 3 on both sides.

    Domain: 0\leq x <2pi

    Given: 3\sin^2{x}= \cos^2{x}

    Divide by \cos^2{x} on both sides: 3\frac{\sin^2{x}}{\cos^2{x}} = 1

    Divide by 3 on both sides: \frac{\sin^2{x}}{\cos^2{x}} = \frac{1}{3}

    Remember that:
    \tan{x} = \frac{\sin{x}}{\cos{x}}

    So let's replace now:

    \tan^2{x} = \frac{1}{3}

    Take the square root:

    \tan{x} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

    x = \arctan{\left(\frac{\sqrt{3}}{3}\right)}

    To find the exact value without using a calculator, use the special 30-60-90 triangle. Check this out for more information:
    Special right triangles - Wikipedia, the free encyclopedia
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  4. #4
    apm
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    thanks everyone
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