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Math Help - sin(x)=1/3. What is cos(x?)

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    sin(x)=1/3. What is cos(x?)

    Given that \sin \theta = \frac{1}{3}, how does one find the exact value of \cos \theta without using a calculator?

    The answer is supposed to be \cos \theta = \frac{2\sqrt{2}}{3}, but I don't see how one reaches it. I know the standard sine/cosine angles ( \frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}...) and their corresponding Cartesian coordinates on the unit circle; is the result supposed to be derived from these? If so, how?
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Draw a right angled triangle.

    Let theta be an angle, 1 will be the opposite side, and 3 will be the hypotenuse.

    Can you find the adjacent side through Pythagoras' Theorem?

    Then, cos of theta is adj/hyp.

    You should get the answer
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  3. #3
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    Use the Pythagorean Identity \displaystyle \sin^2{\theta} + \cos^2{\theta} \equiv 1.
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    Quote Originally Posted by Unknown008 View Post
    Draw a right angled triangle.

    Let theta be an angle, 1 will be the opposite side, and 3 will be the hypotenuse.

    Can you find the adjacent side through Pythagoras' Theorem?

    Then, cos of theta is adj/hyp.

    You should get the answer
    You should get AN answer - there are two possible answers :P
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  5. #5
    MHF Contributor Unknown008's Avatar
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    EDIT: I see what you mean now >.<
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  6. #6
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    Well, you should know that sin^2 x+cos^2x=1. You know sin \theta, cos^2 \theta is... ?

    What do you know about \theta? In which interval is it and how is cos function on that interval (positive or negative)?
    If you know nothing, then you have two solutions.
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  7. #7
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    Ah, thank you all. I knew it was something awfully simple
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