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Math Help - Solving Trig Equations

  1. #1
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    Solving Trig Equations

    Incorporate the use of the double angle formulas identities and other identities to solve for the following on 0 \leq \theta \leq 360

    tanx = sin2x

    My Attempt:
    \frac{sinx}{cosx} = 2sinxcosx

    sinx = 2sinx

    sinx - 2sinx = 0

    -sinx = 0

    x = 0 and 180 and 360


    Textbook Answers:
    0 and 45 and 135 and 180 and 225 and 315 and 360

    What am I doing wrong?
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  2. #2
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    Hello, Macleef!

    A silly mistake . . . don't kick yourself too hard.


    Solve for 0 \leq \theta \leq 360

    \tan x \:= \:\sin2x

    My Attempt:

    \frac{\sin x}{\cos x} \:= \:2\sin x\cos x

    \sin x \:= \:2\sin x . . . . no
    The cosines don't cancel . . .


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

    . . \sin x - 2\sin x\cos^2\!x \:=\:0

    Factor: . \sin x(1 - 2\cos^2\!x) \:=\:0


    Then: . \sin x \:=\:0\quad\Rightarrow\quad \boxed{x \:=\:0^o,\:180^o,\:360^o}

    And: . 1 - 2\cos^2\!x \:=\:0\quad\Rightarrow\quad \cos^2\!x \:=\:\frac{1}{2}
    . . . . \cos x \:=\:\pm\frac{1}{\sqrt{2}}\quad\Rightarrow\quad \boxed{x \;=\;45^o,\:135^o,\:225^o,\:315^o}

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  3. #3
    MHF Contributor
    Joined
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    Quote Originally Posted by Macleef View Post
    Incorporate the use of the double angle formulas identities and other identities to solve for the following on 0 \leq \theta \leq 360

    tanx = sin2x

    My Attempt:
    \frac{sinx}{cosx} = 2sinxcosx

    sinx = 2sinx

    sinx - 2sinx = 0

    -sinx = 0

    x = 0 and 180 and 360


    Textbook Answers:
    0 and 45 and 135 and 180 and 225 and 315 and 360

    What am I doing wrong?
    What did you do wrong?
    From sinX /cosX = 2sinXcosX, how did you get sinX = 2sinX? What happened to the two cosX's?

    You want me to solve the Problem?

    After your sinX/cosX = 2sinXcosX,

    Bring them all to the lefthand side,
    sinX/cosX -2sinXcosX = 0
    sinX(1/cosX -2cosX) = 0

    sinX = 0
    X = arcsin(0) = 0, 180, 360 degrees ---------***

    1/cosX -2cosX = 0
    Clear the fraction, multiply both sides by cosX,
    1 -2cos^2(X) = 0
    1 = 2cos^2(X)
    1/2 = cos^2(X)
    cosX = +,-sqrt(1/2)
    cosX = +,-1/sqrt(2)
    X = arccos(+,-1/sqrt(2))
    X = 45, 135, 225, 315 degrees. ------***

    Therefore, X = 0, 45, 135, 180, 225, 315, 360 degrees.
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