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Thread: trigonometric equation

  1. #1
    Junior Member
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    Unhappy trigonometric equation

    1. Prove that the equation is true.
    cos55*+cos65*+cos175*=0
    2. Solve the equation:
    sin4x+sin2x=sinx
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, blertta!

    Prove: .$\displaystyle \cos55^o+\cos65^o+\cos175^o \:=\:0$
    Sum-to-product identity: .$\displaystyle \cos A + \cos B \:=\:2\cdot\cos\left(\frac{A+B}{2}\right)\cos\left (\frac{A-B}{2}\right)$


    $\displaystyle \text{We have: }\;\cos55^o + \underbrace{\cos65^o + \cos175^o}$

    . . . . .$\displaystyle = \;\cos55^o + \overbrace{2\cdot\cos120^o\cos55^o} $

    . . . . .$\displaystyle = \;\cos55^o + 2\left(-\frac{1}{2}\right)\cos55^o$

    . . . . .$\displaystyle = \;\cos55^o - \cos55^o$

    . . . . .$\displaystyle =\qquad\quad0 $

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  3. #3
    Senior Member
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    Nov 2007
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    $\displaystyle \sin (4x)+\sin (2x)=2\cos x\sin x + 4 \cos^3 x \sin x - 4 \cos x \sin^3 x$.
    So if $\displaystyle \sin x\ne 0$ (You should check $\displaystyle \sin x = 0$ as it might be a solution!): $\displaystyle \cos x \left(1 + 2 \cos^2 x - 2 \sin^2 x\right)=0\iff \cos x\left(1+2\cos 2x\right)=0$.
    So if $\displaystyle \cos x\ne 0$ (You should check $\displaystyle \cos x = 0$ as well!): $\displaystyle \cos 2x=-\frac 12$.
    You can finish it from here. I hope I didn't make any mistakes.
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