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Math Help - Parametric Parameter Elimination

  1. #1
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    Parametric Parameter Elimination

    Eliminate the parameter to find a cartesian equation of the curve.

    x = tan(t)
    y = sec^2(t)

    from 0 <= t <= pi/2

    Thank you!
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  2. #2
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    For eliminating the parameter:

    Get t in terms of x. Then substitute that result into the second equation.

    Alternately, get t in terms of y. Then substitute that result into the first equation.
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  3. #3
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    Quote Originally Posted by Redding1234 View Post
    For eliminating the parameter:

    Get t in terms of x. Then substitute that result into the second equation.

    Alternately, get t in terms of y. Then substitute that result into the first equation.
    y= sec^2(arctan(x)), I've already gotten that, but how does one simplify that?

    Is there some identity that's slipping my mind right now?
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  4. #4
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    You can do it this way: imagine a right triangle with "opposite side" of length x and "near side" of length 1. Then tan(\theta)= x or \theta= arctan x. sec(arctan(x))= sec(\theta) and secant is defined as "hypotenuse over near side". You can get the hypotenuse from the Pythagorean theorem.

    That's really a "visual" way of getting the following identity:

    You know that sin^2(\theta)+ cos^2(\theta)= 1. Divide both sides by cos^2(\theta) to get \frac{sin^2(\theta)}{cos^2(\theta)}+ 1= \frac{1}{cos^2(\theta)} which is the same as tan^2(\theta)+ 1= sec^2(\theta).

    Taking \theta= arctan(x) in that gives gives sec^2(arctan(x))= tan^2(arctan(x)+ 1= x^2+ 1. Therefore, sec(arctan(x))= \sqrt{x^2+ 1}.
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