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Math Help - Given equation Find angles

  1. #1
    Junior Member freestar's Avatar
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    Given equation Find angles

    Which angles \alpha will satisfy the equation \5\sin^2\alpha\ + \25sin\alpha\ - 2.4375 = 0 ?

    I tried the following to no avail.
    \5sin\alpha\(sin\alpha + 1)\ - 2.4375 = 0
    Don't know where to go from here. Maybe use the quadratic formula? Been a while since I've done these.

    Thanks
    Last edited by freestar; August 31st 2011 at 08:13 PM.
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  2. #2
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    Re: Given equation Find angles

    ?? I see you have considered the Quadratic Formula? Do it! \sin(\alpha) = ??
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    Junior Member freestar's Avatar
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    Re: Given equation Find angles

    Thanks TK, I did the quadratic formula:
     \frac{\-25\pm\sqrt{625 + 20(2.4375)}}{10}

    Took the acrsin of 0.09... Still not getting it.
    Last edited by freestar; August 31st 2011 at 08:32 PM.
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  4. #4
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    Re: Given equation Find angles

    What are you not getting? That looks good. You threw out -5.095669... That was right on track.

    arcsin is good, but not quite all the answer. There is a twin over there around pi!

    Note sin(x) = x is a very good approximation for very small x. Don't be too freaked out when you hit the arcsin button and for a moment you think nothing happened.
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  5. #5
    Junior Member freestar's Avatar
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    Re: Given equation Find angles

    Apologies, I didn't understand when you said There is a twin over there around pie. So the answer would be 1.7pi? I'm thinking \sin\alpha=0.09..Sorry for being an idiot.
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Given equation Find angles

    In general if you have the goniometric equation:
    \sin(x)=a
    Two solutions:
    \Leftrightarrow x=\arcsin(a)+2k\pi
    \Leftrightarrow x=\pi - \arcsin(a)+2k\pi
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  7. #7
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    Re: Given equation Find angles

    Quote Originally Posted by freestar View Post
    Which angles \alpha will satisfy the equation \5\sin^2\alpha\ + \25sin\alpha\ - 2.4375 = 0 ?

    I tried the following to no avail.
    \5sin\alpha\(sin\alpha + 1)\ - 2.4375 = 0
    Don't know where to go from here. Maybe use the quadratic formula? Been a while since I've done these.

    Thanks
    Let \displaystyle x = \sin{\alpha} so that your equation becomes \displaystyle 5x^2 + 25x - 2.4375 = 0. Solve for \displaystyle x and use this to solve for \displaystyle \alpha.
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  8. #8
    Junior Member freestar's Avatar
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    Re: Given equation Find angles

    I replaced \sin\alpha with x
    5x^2 + 25x - 2.4375 = 0
    x^2 + x - 0.4875 = 0
    \frac{-1\pm\sqrt{1+4(0.4875)}}{2}
    solving for \sin\alpha
    \sin^-^1(0.35877) = 21.0247degrees
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  9. #9
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    Re: Given equation Find angles

    Quote Originally Posted by freestar View Post
    I replaced \sin\alpha with x
    5x^2 + 25x - 2.4375 = 0
    x^2 + x - 0.4875 = 0
    \frac{-1\pm\sqrt{1+4(0.4875)}}{2}
    solving for \sin\alpha
    \sin^-^1(0.35877) = 21.0247degrees
    When you divide everything by 5 you should get \displaystyle x^2 + 5x - 0.4875 = 0.
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  10. #10
    Junior Member freestar's Avatar
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    Re: Given equation Find angles

    Sorry. Silly mistake!
    Correction:

    x^2 + 5x - 0.4875 = 0
    \frac{-5\pm\sqrt{25+4(0.4875)}}{2}
    solving for \sin\alpha
    \sin^-^1(0.09567) = 5.4899degrees
    What would be next?
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    Re: Given equation Find angles

    Quote Originally Posted by freestar View Post
    Sorry. Silly mistake!
    Correction:

    x^2 + 5x - 0.4875 = 0
    \frac{-5\pm\sqrt{25+4(0.4875)}}{2}
    solving for \sin\alpha
    \sin^-^1(0.09567) = 5.4899degrees
    What would be next?
    You are probably expected to give the answer in radians, but anyway. You have the answer that corresponds to the first quadrant. Where else is the sine function positive? Where does the sine function repeat itself?
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  12. #12
    Junior Member freestar's Avatar
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    Re: Given equation Find angles

    Yes sir, I want to find the answer in pi and that is what I am having difficulty figuring out because when I try to convert it, I just get some decimal number times pi.
    The other angle would be 180-5.4899 = 174.5 degrees. Sine is positive in the top right and left (1st and 2nd quadrant).
    How can I write them as proper pi values? Really appreciate the help eh.
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  13. #13
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    Re: Given equation Find angles

    Quote Originally Posted by freestar View Post
    Yes sir, I want to find the answer in pi and that is what I am having difficulty figuring out because when I try to convert it, I just get some decimal number times pi.
    The other angle would be 180-5.4899 = 174.5 degrees. Sine is positive in the top right and left (1st and 2nd quadrant).
    How can I write them as proper pi values? Really appreciate the help eh.
    You should be well aware that \displaystyle 2\pi^C = 360^{\circ} \implies \frac{2\pi}{360}^C = 1^{\circ} \implies 1^{\circ} = \frac{\pi}{180}^C.

    So to convert degrees to radians, you need to multiply by \displaystyle \frac{\pi}{180}. Since you have decimal approximations for your degrees, I expect you don't need to leave your radian answers in terms of \displaystyle \pi.
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  14. #14
    Junior Member freestar's Avatar
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    Re: Given equation Find angles

    I know how to convert them to radians. I guess I was just trying to figure out if I could get it to be a nice fraction time pi or something. I'll just leave the radians in decimals. Thanks a lot! Problem solved.
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