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Math Help - Algebraic Expressions in Trig.

  1. #1
    Junior Member Godfather's Avatar
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    Algebraic Expressions in Trig.

    Find an algebraic expression equivalent to tan ( arc cos X )


    Evaluate:
    arc cos ( cos pi/3)
    arc cos ( cos -pi/7)
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by Godfather View Post
    Find an algebraic expression equivalent to tan ( arc cos X )


    Evaluate:
    arc cos ( cos pi/3)
    arc cos ( cos -pi/7)
    Draw a reference triangle using the defintion of the cosine function...

    1 will be the hypotenuse and x will be the adjacent side....

    solve for the opposite side using the pythagorean theorem.

    x^2+(opp)^2=1^2

    opp=\sqrt{1-x^2}

    Using the definition of tangent gives

    tan(\theta)=\frac{opp}{adj}=\frac{\sqrt{1-x^2}}{x}
    Last edited by TheEmptySet; February 24th 2008 at 10:00 AM. Reason: I hit the wrong button
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    For the second part....

    We need to know the domain of the arc cosine function....

    The domain is 0 to Pi.

    also note that the cosine function is even ie

    cos(-\theta)=cos(\theta)

    This gives us

    cos\left(\frac{-\pi}{7}\right)=cos\left(\frac{\pi}{7}\right)

    so

    cos^{-1}(cos\left(\frac{\pi}{3}\right))=\frac{\pi}{3}

    and

    cos^{-1}(cos\left(\frac{-\pi}{7}\right))=cos^{-1}(cos\left(\frac{\pi}{7}\right))=\frac{\pi}{7}

    since the angle is now in the domain of the arc sine function.
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