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Math Help - General solution

  1. #1
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    General solution

    Write general solutions for these trig equations

    sin3a+sina = 0

    cos4x+cos2x = 0

    any help greatly appreciated. thanx !!
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  2. #2
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    Hello, THSKluv!


    We need two Sum-to-Product Identities:

    . . \begin{array}{cccc}{\color{blue}(1)} &\sin A + \sin B &=& 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\ \\ {\color{blue}(2)} & \cos A + \cos B &=& 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \end{array}



    Write general solutions for these trig equations:

    . . \sin3a+\sin a \:=\: 0
    Using [1], we have: . 2\sin(2a)\cos(a) \:=\:0


    Then: . \begin{array}{ccccc}\sin(2a) \:=\:0 & \Rightarrow& 2a\:=\:\pi n &\Rightarrow& \boxed{a \:=\:\frac{\pi}{2}n} \\<br />
\cos(a) \:=\:0 &\Rightarrow& \boxed{a \:=\:\frac{\pi}{2} + \pi n} \end{array}




    \cos4x+\cos2x \:=\: 0
    Using [2], we have: . 2\cos(3x)\cos(x) \:=\:0

    Then: . \begin{array}{ccccc}\cos(3x)\:=\:0 &\Rightarrow& 3x \:=\:\frac{\pi}{2} + \pi n &\Rightarrow& \boxed{x \:=\:\frac{\pi}{6} + \frac{\pi}{3}n} \\ <br />
\cos(x) \:=\:0 &\Rightarrow& \boxed{x \:=\:\frac{\pi}{2} + \pi n} \end{array}

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