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Math Help - Rearranging equation

  1. #1
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    Rearranging equation

    Can you rearrange an equation like what i have done?

     a^2cos^2\theta a^2sin^2\theta d\theta  = a^4cos^2\theta sin^2\theta d\theta
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  2. #2
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    Quote Originally Posted by Karl Harder View Post
    Can you rearrange an equation like what i have done?

     a^2cos^2\theta a^2sin^2\theta d\theta  = a^4cos^2\theta sin^2\theta d\theta
    as long as each a is a distinct constant factor, you sure can.

    you can also do this ...

    a^4\cos^2{\theta} \sin^2{\theta} \, d\theta =

    \frac{a^4}{4} \sin^2(2\theta) \, d\theta =

    \frac{a^4}{8}[1 - \cos(4\theta)] \, d\theta
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  3. #3
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    Quote Originally Posted by skeeter View Post
    as long as each a is a distinct constant factor, you sure can.

    you can also do this ...

    a^4\cos^2{\theta} \sin^2{\theta} \, d\theta =

    \frac{a^4}{4} \sin^2(2\theta) \, d\theta =

    \frac{a^4}{8}[1 - \cos(4\theta)] \, d\theta
    What maths principles did you use to get these answers. I have been trying to use the half angle formulas is this correct?
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  4. #4
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    Quote Originally Posted by Karl Harder View Post
    What maths principles did you use to get these answers. I have been trying to use the half angle formulas is this correct?
    I used a double angle formula and a power reduction identity (derived from a double angle formula) ...

    \sin(2x) = 2\sin{x}\cos{x}

    \sin^2{u} = \frac{1 - \cos(2u)}{2}
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  5. #5
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    I'm still stuck on how you get this using the double angle formula?

    a^4\cos^2{\theta} \sin^2{\theta} \, d\theta = \frac{a^4}{4} \sin^2(2\theta) \, d\theta
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  6. #6
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    Quote Originally Posted by Karl Harder View Post
    I'm still stuck on how you get this using the double angle formula?

    a^4\cos^2{\theta} \sin^2{\theta} \, d\theta = \frac{a^4}{4} \sin^2(2\theta) \, d\theta
    never mind i have found out that i can get the answer by using the proudct formula
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  7. #7
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    Quote Originally Posted by Karl Harder View Post
    I'm still stuck on how you get this using the double angle formula?

    a^4\cos^2{\theta} \sin^2{\theta} \, d\theta = \frac{a^4}{4} \sin^2(2\theta) \, d\theta


    ... never mind i have found out that i can get the answer by using the proudct formula
    product formula? ...

    a^4 \cos^2{\theta} \sin^2{\theta}

    \frac{a^4}{4} \cdot 4\cos^2{\theta} \sin^2{\theta}

    \frac{a^4}{4} (2\cos{\theta} \sin{\theta})^2

    \frac{a^4}{4} \sin^2(2\theta)
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