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Thread: Trig problem w/ answer - help me understand this answer

  1. #1
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    Trig problem w/ answer - help me understand this answer

    Hi,

    PROBLEM: "Find all values of x such that sin2x = sinx and 0 ≤ x ≤ 2π" (that last symbol is "Pi" in case it's hard to read)

    ANSWERS: 0, π/3, π, 5π/3, 2π


    MY THOUGHT PROCESS BEFORE SEEING THE ANSWER WAS:

    (1) double-angle formula: sin2x = (2)(sinx)(cosx)
    (2) sin2x = sinx only when (2)(cosx) = 1
    (3) (2)(cosx) = 1 only when cosx = 1/2
    (4) cosx = 1/2 only when theta = π/3

    Yes, π/3 is an answer, yet there are more answers (0, π, 5π/3, 2π). So, what was wrong with my thought process, and why did it lead me to only one out of the 5 answers?

    And what would be a better way to approach this problem, or a way to fix my initial approach?

    Thanks.
    Last edited by BYUguy; Oct 18th 2015 at 05:43 PM.
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  2. #2
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    Re: Trig problem w/ answer - help me understand this answer

    $\sin(2x)=\sin{x}$

    $2\sin{x}\cos{x}=\sin{x}$

    $2\sin{x}\cos{x}-\sin{x}=0$

    $\sin{x}(2\cos{x}-1)=0$

    set each factor equal to zero and solve ...

    $\sin{x}=0$ at $x=0$, $x=\pi$, and $x=2\pi$

    $2\cos{x}-1=0 \implies \cos{x}=\dfrac{1}{2}$.

    Last two solutions are $x=\dfrac{\pi}{3}$ and $x=\dfrac{5\pi}{3}$
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  3. #3
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    Re: Trig problem w/ answer - help me understand this answer

    Oh, yeah, now I see that even the way I was looking at it should have given me 5π/3 right away. And actually the other three as well.

    Thanks!
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  4. #4
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    Re: Trig problem w/ answer - help me understand this answer

    Quote Originally Posted by BYUguy View Post
    (2) sin2x = sinx only when (2)(cosx) = 1
    THIS is your mistake. The cardinal rule of numbers is that you can not divide by 0. As Skeeter has shown you, it is possible that sin(x) = 0 (in fact, it's where the other solutions come from).

    ALWAYS solve equations by using the Null Factor Law if possible.
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  5. #5
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    Re: Trig problem w/ answer - help me understand this answer

    Thanks to both of you!
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