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Math Help - solving trig equations

  1. #1
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    solving trig equations

    Solve each equation. 0 < x < 2pi
    1. 3sinx = cosx
    2. sin2x = sinx

    Find the points of intersection of each pair of curves in the given interval.
    y=sin2x, y=sinx
    0 < x < 2pi
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  2. #2
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    Solve each equation. . 0 \leq x \leq 2\pi

    1)\;\;3\sin x \:= \:\cos x

    Divide both sides by \cos x\!:\;\;\frac{3\sin x}{\cos x} \:=\:\frac{\cos x}{\cos x}\quad\Rightarrow\quad 3\tan x \:=\:1

    Then: . \tan x\:=\:\frac{1}{3}\quad\Rightarrow\quad x \:=\:\tan^{-1}\left(\frac{1}{3}\right)\quad\Rightarrow\quad\bo  xed{ x\;\approx\;0.32175,\;3.46334}



    2)\;\;\sin2x \:= \:\sin x

    We have: . 2\sin x\cos x \:=\:\sin x

    . . 2\sin x\cos x - \sin x \:=\:0\quad\Rightarrow\quad \sin x(2\cos x - 1)\:=\:0

    Then:. . \sin x \:=\: 0 \quad\Rightarrow\quad\boxed{ x \:=\:0,\:\pi,\:2\pi}
    . . . . . . 2\cos x - 1 \:= \:0 \quad\Rightarrow\quad \cos x \:=\: \frac{1}{2} \quad\Rightarrow\quad \boxed{x\:=\: \frac{\pi}{3},\;\frac{5\pi}{3}}



    Find the points of intersection of this pair of curves in the given interval.

    . . \begin{array}{ccc}y &=&\sin2x \\y &= &\sin x\end{array}\qquad 0\,<\,x\,<\,2\pi

    This has the same equation as the preceding problem.

    Answers: . \boxed{x \:=\:\pi,\;\frac{\pi}{3},\;\frac{5\pi}{3}}

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