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Math Help - Trig general solution

  1. #1
    Member srirahulan's Avatar
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    Lightbulb Trig general solution

    cos5x+cosx-sin4x=0

    Do you have any idea to simplify and find a general solution for it.
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    Re: Trig general solution

    Quote Originally Posted by srirahulan View Post
    cos5x+cosx-sin4x=0

    Do you have any idea to simplify and find a general solution for it.
    Try to use angle sum and double angle identities, with the Pythagorean Identity, to reduce this to a polynomial function in sin(x) or cos(x).
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  3. #3
    Member srirahulan's Avatar
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    Re: Trig general solution

    I can't understand your solution.
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    Re: Trig general solution

    Quote Originally Posted by srirahulan View Post
    I can't understand your solution.
    You could TRY doing what you were instructed. Then I might be able to guide you some more...
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  5. #5
    Member srirahulan's Avatar
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    Re: Trig general solution

    cos5x+cosx-sin4x=0\\2cos3xcos2x-2sin2xcos2x=0\\2cos2x(cos3x-sin2x)=0\\\mbox{Then how can i eliminate sin2x.}
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  6. #6
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    Re: Trig general solution

    Quote Originally Posted by srirahulan View Post
    cos5x+cosx-sin4x=0\\2cos3xcos2x-2sin2xcos2x=0\\2cos2x(cos3x-sin2x)=0\\\mbox{Then how can i eliminate sin2x.}
    First of all, \displaystyle \begin{align*} \cos{5x} \end{align*} is not \displaystyle \begin{align*} 2\cos{3x}\cos{2x} \end{align*}. If you split it up as \displaystyle \begin{align*} \cos{(3x + 2x)} \end{align*} you will need to use this identity: \displaystyle \begin{align*} \cos{(\alpha + \beta )} \equiv \cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta} \end{align*}. This is the angle sum identity I mentioned in my first post.

    It also appears that the \displaystyle \begin{align*} \cos{x} \end{align*} term has disappeared in the second line.
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  7. #7
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Trig general solution

    Quote Originally Posted by srirahulan View Post
    f(x)=cos5x+cosx-sin4x=0
    Do you have any idea to simplify and find a general solution for it.
    Let z=x+i y be a complex number. Substitute cos(5x), cos(x), sin(4x) with z component and solve for z...then arguments of z solutions are solutions to f(x) as follows:

    f(z)=\frac{1}{2}\left(z^5+\frac{1}{z^5} \right)+ \frac{1}{2} \left(z+\frac{1}{z}\right)-\frac{1}{2i}\left(z^4-\frac{1}{z^4}\right)

    f(z) simplifies to:

    f(z)=\frac{(-i+z) (i+z) \left(-i+z^2\right) \left(i+z^2\right) \left(1-i z-z^2+i z^3+z^4\right)}{2 z^5} =0 z is not 0

    z= \left\{-i,i,-(-1)^{1/4},(-1)^{1/4},-(-1)^{3/4},(-1)^{3/4},-\frac{1}{4} i \left(1+\sqrt{5}-i \sqrt{2 \left(5-\sqrt{5}\right)}\right),-\frac{1}{4} i \left(1+\sqrt{5}+i \sqrt{2 \left(5-\sqrt{5}\right)}\right),\frac{1}{4} i \left(-1+\sqrt{5}-i \sqrt{2 \left(5+\sqrt{5}\right)}\right),\frac{1}{4} i \left(-1+\sqrt{5}+i \sqrt{2 \left(5+\sqrt{5}\right)}\right)\right\}

    Arguments of z are as follows:

    x=\left\{-\frac{\pi }{2},\frac{\pi }{2},-\frac{3 \pi }{4},\frac{\pi }{4},-\frac{\pi }{4},\frac{3 \pi }{4},-\pi -\text{ArcTan}\left[\frac{-1-\sqrt{5}}{\sqrt{2 \left(5-\sqrt{5}\right)}}\right],\text{ArcTan}\left[\frac{-1-\sqrt{5}}{\sqrt{2 \left(5-\sqrt{5}\right)}}\right],\text{ArcTan}\left[\frac{-1+\sqrt{5}}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right],\pi -\text{ArcTan}\left[\frac{-1+\sqrt{5}}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right]\right\}
    Last edited by MaxJasper; August 31st 2012 at 10:57 AM.
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    Re: Trig general solution

     \cos A+\cos B=2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)
    so

    \cos 5x+\cos x=2\cos 3x\cos 2x.

    That allows the equation to be written as

    2\cos 2x(\cos 3x - \sin 2x)=0.

    Now use the identities

    \cos 3x = 4\cos^{3}x-3\cos x and \sin 2x = 2\sin x \cos x,

    remove \cos x as a factor and turn what's left inside the brackets into a quadratic in \sin x.
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  9. #9
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    Re: Trig general solution

    Quote Originally Posted by srirahulan View Post
    cos5x+cosx-sin4x=0\\2cos3xcos2x-2sin2xcos2x=0\\2cos2x(cos3x-sin2x)=0\\\mbox{Then how can i eliminate sin2x.}
    I apologise for my previous post, you did use a correct identity here.
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