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Math Help - trig equations

  1. #1
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    trig equations

    hi guys , i am a high school student , and i have a trigonometry test tomorrow . i would appreciate every help from you.
    for now i can't solve these trigonometry equations.
    they are mashing up my mind


    1) sin2x + sinx * cosx = 1

    2)1+cosx + cos(x/2) =0

    3) 1 + cos(π +x ) + cos(π/2 + x/2) =0
    Last edited by enea54; December 5th 2012 at 08:18 AM.
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  2. #2
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    Re: trig equations

    So you just want someone else to do them for you?
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  3. #3
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    Re: trig equations

    Quote Originally Posted by HallsofIvy View Post
    So you just want someone else to do them for you?
    I am doing a whole summary of the trigonometry chapter at the moment , so that i'll be more prepared for tha exam. These 3 equations were the only ones i couldn't solve , i asked help from my parents , my brother but they couldn't solve it either. I wasn't left with much choices so i went online and posted them here and i think this is the most adequate place to do it. No, i'm not asking someone else to do them for me . I just want a hint, a way of doing them so that i can learn something and put it to use later.
    Last edited by enea54; December 5th 2012 at 08:15 AM.
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  4. #4
    Forum Admin topsquark's Avatar
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    Re: trig equations

    Quote Originally Posted by enea54 View Post
    1) sin2x + sinx * cosx = 1
    Start with
    sin^2(x) + sin(x)~cos(x) = 1

    (sin^2(x) -1) + sin(x)~cos(x) = 0

    -cos^2(x) + sin(x)~cos(x) = 0

    I'd multiply both sides by -1, but you can do it from this too. Factor out the common cos(x) and the solution follows easily from there.

    -Dan
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  5. #5
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    Re: trig equations

    thanks a lot , that was what i needed, thanks
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  6. #6
    MHF Contributor ebaines's Avatar
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    Re: trig equations

    For the second one you can replace 'x' with a new variable 'w' where x = 2w. This gives

    1 + \cos(2w) + \cos(w) = 0

    Now use the identity cos(2w)= 2cos^2w-1 and solve for cos(w), and from that determine values for w, then x.
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  7. #7
    Forum Admin topsquark's Avatar
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    Re: trig equations

    Ya beat me ebaines! Almost word for word.

    -Dan
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  8. #8
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    Re: trig equations

    Notice that sin^2(x) + sin(x)~cos(x) = 1

    can be written as \cos(x)[\sin(x)-\cos(x)]=0
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  9. #9
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    Re: trig equations

    Hello, enea54!

    Here's the last one . . .


    (3)\;1 + \cos(x + \pi) + \cos\left(\tfrac{x}{2}+\tfrac{\pi}{2}\right) \:=\:0

    We have: . 1 + \cos(x + \pi) + \cos\left(\tfrac{x+\pi}{2}\right) \:=\:0

    Let \theta \:=\:x + \pi

    We have: . 1 + \cos\theta + \cos\tfrac{\theta}{2} \:=\:0

    n . 1 + \left(2\cos^2\!\tfrac{\theta}{2} - 1\right) + \cos\tfrac{\theta}{2} \:=\:0

    . . . . . . . . . 2\cos^2\!\tfrac{\theta}{2} + \cos\tfrac{\theta}{2} \:=\:0

    n . . . . . \cos\tfrac{\theta}{2}\left(2\cos\tfrac{\theta}{2} + 1\right) \:=\:0


    \cos\tfrac{\theta}{2} \:=\:0 \quad\Rightarrow\quad \tfrac{\theta}{2} \:=\:\begin{Bmatrix}\frac{\pi}{2} \\ \\[-4mm] \frac{3\pi}{2} \end{Bmatrix} \quad\Rightarrow\quad \theta \:=\:\begin{Bmatrix} \pi \\ 3\pi \end{Bmatrix}

    . . . . x + \pi \:=\:\begin{Bmatrix}\pi \\ 3\pi\end{Bmatrix} \quad\Rightarrow\quad \boxed{x \:=\:\begin{Bmatrix}0 \\ 2\pi\end{Bmatrix}}


    2\cos\tfrac{\theta}{2} + 1 \:=\:0 \quad\Rightarrow\quad \cos\tfrac{\theta}{2} \:=\:-\tfrac{1}{2} \quad\Rightarrow\quad \tfrac{\theta}{2} \:=\:\begin{Bmatrix}\frac{2\pi}{3} \\ \\[-4mm] \frac{4\pi}{3} \end{Bmatrix} \quad\Rightarrow\quad \theta \:=\:\begin{Bmatrix}\frac{4\pi}{3} \\ \\[-4mm] \frac{8\pi}{3} \end{Bmatrix}

    . . . . x + \pi \:=\:\begin{Bmatrix}\frac{4\pi}{3} \\ \\[-4mm] \frac{8\pi}{3} \end{Bmatrix} \quad\Rightarrow\quad \boxed{x \:=\:\begin{Bmatrix} \frac{\pi}{3} \\ \\[-4mm] \frac{5\pi}{3} \end{Bmatrix}}

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  10. #10
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    Re: trig equations

    thanks a lot guys, you are the best of the best , thanks again
    Last edited by enea54; December 5th 2012 at 11:03 AM.
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