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Math Help - Simplify of cosinus and sinus terms

  1. #1
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    Simplify of cosinus and sinus terms

    Hi,

    I have a calculation where I have to simplify some trogonometric terms. I did it up to here und and to here I am sure that it es correct but now I stumble. How to go on?

    Target of the calculation is: r^2 \sin(\Theta)

    This is the point I stumble:

    r^2 [ \cos(\Theta)^2 \sin(\Theta) \cos(\varphi)^2 + \sin(\Theta)^3 \sin(\varphi)^2 + \cos(\Theta)^2 \sin(\varphi)^2 \sin(\Theta) + \sin(\Theta)^3 \cos(\varphi)^2 ]

    Whats next?

    I guess I have to use that sin square plus cos square is one, but I guess here it would be wrong.

    thanks
    greetings
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  2. #2
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    Quote Originally Posted by Herbststurm View Post
    Hi,

    I have a calculation where I have to simplify some trogonometric terms. I did it up to here und and to here I am sure that it es correct but now I stumble. How to go on?

    Target of the calculation is: r^2 \sin(\Theta)

    This is the point I stumble:

    r^2 [ \cos(\Theta)^2 \sin(\Theta) \cos(\varphi)^2 + \sin(\Theta)^3 \sin(\varphi)^2 + \cos(\Theta)^2 \sin(\varphi)^2 \sin(\Theta) + \sin(\Theta)^3 \cos(\varphi)^2 ]

    Whats next?

    I guess I have to use that sin square plus cos square is one, but I guess here it would be wrong.

    thanks
    greetings
    Take out a factor of  \sin{\Theta}

    r^2\sin{\Theta} [ \cos(\Theta)^2 \cos(\varphi)^2 + \sin(\Theta)^2 \sin(\varphi)^2 + \cos(\Theta)^2 \sin(\varphi)^2  + \sin(\Theta)^2 \cos(\varphi)^2 ]


    r^2\sin{\Theta} [ (\cos(\Theta) \cos(\varphi))^2 + (\sin(\Theta) \sin(\varphi))^2 + (\cos(\Theta) \sin(\varphi))^2  + (\sin(\Theta) \cos(\varphi))^2 ]

    Can you now simplify the rest of the terms in the bracket to 1?

    You can use the following to change the terms left over:

    \cos(\Theta \pm \varphi) = \cos(\Theta)\cos(\varphi) \mp \sin(\Theta)\sin(\varphi)

    \sin(\Theta \pm \varphi) = \sin(\Theta)\cos(\varphi) \pm \cos(\Theta)\sin(\varphi)

    To use these to find identities for the terms you have, you have to add or subtract these equations from each other, depending on what you want.
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  3. #3
    MHF Contributor red_dog's Avatar
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    \cos^2\theta\sin\theta\cos^2\phi+\sin^3\theta\sin^  2\phi+\cos^2\theta\sin^2\phi\sin\theta+\sin^3\thet  a\cos^2\phi=

    =\cos^2\theta\sin\theta(\cos^2\phi+\sin^2\phi)+\si  n^3\theta(\sin^2\phi+\cos^2\phi)=

    =\cos^2\theta\sin\theta+\sin^3\theta=\sin\theta(\c  os^2\theta+\sin^2\theta)=\sin\theta
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  4. #4
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    Great, now I got it

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