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Math Help - How to find point of interesection between sin x and sin 0.5x?

  1. #1
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    Cool How to find point of interesection between sin x and sin 0.5x?

    I'm having trouble manually working out points of intersection between trig graphs. I can do them when its between a cos graph and a sin graph using cos^2(x) + sin^2(x) = 1 but I cant do them when they are the same etc. sin x = sin 2x

    How do you manually work these pts of intersection out? Do I have to use the general formula?
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  2. #2
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    Re: How to find point of interesection between sin x and sin 0.5x?

    You can use the fact that sin(2x) - sin(x) = 2sin(x/2)cos(3x/2).
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  3. #3
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    Re: How to find point of interesection between sin x and sin 0.5x?

    Hello, sorkii!

    \text{How to find the intersections of }\,y \:=\:\sin x\,\text{ and }\,y \,=\,\sin\tfrac{x}{2}

    We wish to solve: . \sin x \:=\:\sin\tfrac{x}{2}


    Recall the identity: . \sin\theta \:=\:2\sin\tfrac{\theta}{2}\cos\tfrac{\theta}{2}


    So we have: . 2\sin\tfrac{x}{2}\cos\tfrac{x}{2} \:=\:\sin\tfrac{x}{2} \quad\Rightarrow\quad 2\sin\tfrac{x}{2}\cos\tfrac{x}{2} - \sin\tfrac{x}{2} \;=\;0

    . . . . . . \sin\tfrac{x}{2}\big[2\cos\tfrac{x}{2} - 1\big] \;=\;0


    Hence: . \sin\tfrac{x}{2} \:=\:0 \quad\Rightarrow\quad \tfrac{x}{2} \:=\:\pi n \quad\Rightarrow\quad \boxed{x \:=\:2\pi n}

    And: . 2\cos\tfrac{x}{2}-1 \:=\:0 \quad\Rightarrow\quad  \cos\tfrac{x}{2} \:=\:\tfrac{1}{2}

    . . . . . \tfrac{x}{2} \:=\:\pm\tfrac{\pi}{3}+ 2\pi n \quad\Rightarrow\quad \boxed{x \:=\: \pm\tfrac{2\pi}{3} + 4\pi n}

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