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Find the angle made by two tangents

  1. Mar 1, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the angle made by the two tangents to the curve ##x=\sin2t## and ##y=\sin(2t)\cos(2t)## at the point ##(0,0)##

    2. Relevant equations
    ##\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##
    (Derivative of a parametric equation)

    3. The attempt at a solution
    ## x = \sin(2t) = 0 ## when ## t = 0, \frac{\pi}{2}, \pi,... ##
    ## y=\sin(2t)\cos(2t) = 0 ## at the same values of ## t##
    Taking the derivative of the parametric equations by using the formula in part two, I get ##\frac{2\cos^2(2t)-2t\sin^2t}{2\cos(2t)}## I get to plug in any value of t, so I choose ##t=\pi/2##
    With that t value, I get ## \frac{2(-1^2)-2(0)}{-2}## which is equal to ##-1##
    Now I'll try ##t=0##, and I get ##1##. Using ##t=\pi## I also get one. Therefore, the angle must be between -1 and 1 and be equal to ## t = 0, \frac{\pi}{2}, \pi,... ##
    My answer is 0.
    The correct answer is ##\frac{\pi}{2}##
    Could someone please enlighten me as to my mistake? That would be highly appreciated. Thanks.
     
    Last edited by a moderator: Mar 1, 2015
  2. jcsd
  3. Mar 1, 2015 #2
    Think about what you calculated...
    The slope you got at one time was 1, the slope you got at another time was -1 (both at the same point).
    What is the angle between a line of slope 1 and a line of slope -1?

    You don't want to guess that because you got -1 and 1 as answers, the best thing to do is average them to get 0. -1 and 1 have nothing directly to do with angles, those are your slopes (dy/dx)
     
  4. Mar 1, 2015 #3
    I see! The angle between them is 90 degrees. Thank you so much.
     
  5. Mar 1, 2015 #4
    Now I get it! The angle is 90 degrees. Thank you so much!
     
  6. Mar 1, 2015 #5
    Glad to help. and just so you can visualize the parametrization:
    http://www4b.wolframalpha.com/Calculate/MSP/MSP11162061a3322c6679i6000012f58ad84fgbh939?MSPStoreType=image/gif&s=64&w=286.&h=97.&cdf=Animation [Broken]
    We have two lines that cross the origin, with the slopes that you figured out.
     
    Last edited by a moderator: May 7, 2017
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