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Math Help - Finding sinx formed by 2 rays.

  1. #1
    Senior Member sakonpure6's Avatar
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    Finding sinx formed by 2 rays.

    Hello, can some one please check my work, I believe I did it correctly!Finding sinx formed by 2 rays.-sinx.jpg

    Also can you please check if I did Angular velocity correct.

    (Sorry for bothering you >.<! I have a test on Tuesday!)
    Attached Thumbnails Attached Thumbnails Finding sinx formed by 2 rays.-angular-v.jpg  
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  2. #2
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    Re: Finding sinx formed by 2 rays.

    You did both correctly.

    For the first part, you can figure out most if it using only geometry. A right triangle with two equal sides is a 45-45-90 triangle. A right triangle with sides x, x\sqrt{3} is a 30-60-90 triangle (where \theta_2 is the 30 degree angle). So, \theta_1 = \dfrac{\pi}{4}, \theta_2 = \dfrac{\pi}{6}. This means \theta = \pi - \dfrac{\pi}{4} - \dfrac{\pi}{6} = \dfrac{7\pi}{12}.

    You get the same answer either way.
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    Senior Member sakonpure6's Avatar
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    Re: Finding sinx formed by 2 rays.

    Thanks! !
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    Re: Finding sinx formed by 2 rays.

    Quote Originally Posted by SlipEternal View Post
    This means \theta = \pi - \dfrac{\pi}{4} - \dfrac{\pi}{6} = \dfrac{7\pi}{12}.

    You get the same answer either way.
    Yes, but would you figure out that \sin(7\pi/12)=\frac{\sqrt{2}(1+\sqrt{3})}{4}? It looks like this problem asked to find the exact value of \sin\theta. It is probably not a coincidence that both vectors have integer lengths.
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    Senior Member sakonpure6's Avatar
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    Re: Finding sinx formed by 2 rays.

    Can you please explain how a triangle with x, xroot3 yields a 30 , 60, 90 triangle and how yoh figured out that the other angle is pi /4
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    Re: Finding sinx formed by 2 rays.

    I was taught that the sides of a 30-60-90 triangle will have lengths x, x\sqrt{3}, 2x where x is the length of the side opposite the 30 degree angle, x\sqrt{3} is the length of the side opposite the 60 degree angle, and 2x is the length of the hypotenuse in geometry class. I assumed it was still taught. As for the other angle, convert degrees to radians: 45\dfrac{\pi}{180} = \dfrac{\pi}{4}.
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  7. #7
    Senior Member sakonpure6's Avatar
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    Re: Finding sinx formed by 2 rays.

    Oh I see, its the special triangle. Thanks!
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    Re: Finding sinx formed by 2 rays.

    Quote Originally Posted by emakarov View Post
    Yes, but would you figure out that \sin(7\pi/12)=\frac{\sqrt{2}(1+\sqrt{3})}{4}? It looks like this problem asked to find the exact value of \sin\theta. It is probably not a coincidence that both vectors have integer lengths.
    I got a different result when applying the half-angle formula. I got:

    \sin\left(\dfrac{7\pi}{12}\right) = \sin\left(\dfrac{\pi}{2}\right)\cos\left(\dfrac{ \pi}{12}\right) + \sin\left(\dfrac{\pi}{12}\right) \cos\left(\dfrac{\pi}{2}\right) = \cos\left(\dfrac{\pi}{12}\right).

    Since \cos(2\theta) = 2\cos^2\theta-1, the half-angle formula gives:

    \cos \theta = \sqrt{\dfrac{\cos(2\theta)+1}{2}}

    Plugging in \theta = \dfrac{\pi}{12} you have

    \cos \left( \dfrac{\pi}{12}\right) = \sqrt{\dfrac{\cos\left(\dfrac{\pi}{6}\right)+1}{2}  } = \sqrt{\dfrac{\dfrac{\sqrt{3}}{2}+1}{2}} = \dfrac{1}{2}\sqrt{\sqrt{3}+2}

    So, I got a different result.

    Edit: I just checked wolframalpha. Apparently, they are the same.
    Last edited by SlipEternal; November 10th 2013 at 01:01 PM.
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    Re: Finding sinx formed by 2 rays.

    I calculated mine using vector product: \sin\theta=\frac{|u\times v|}{|u|\cdot|v|}. It's also possible to use dot product to find cosine first. If the angles are arbitrary, I would guess the problem authors' intention was to use the formula

    \sin(\pi-\theta_1-\theta_2)= \sin(\theta_1+\theta_2)=\sin(\theta_1)\cos(\theta_  2)+ \cos(\theta_1)\sin(\theta_2)

    The values in the right-hand side can be found by definition.
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