1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Deriving equation for circle using sin and cos identities

  1. Sep 17, 2008 #1
    1. The problem statement, all variables and given/known data
    I have been given these two equations:
    x=2acos^2(x) , y = 2a(cos(x))(sin(x)) where a ranges from 0 to 5 and -2π < x < 2π
    I need to prove that these equations (when you plug in values for x) create points that when plotted, give you a circle with center (x-a) and radius a.

    2. Relevant equations
    I have gotten the correct answer, however I am missing a step and I'm not sure what I did to get the correct answer.

    3. The attempt at a solution
    I wanted to get an equation for a circle out of the two equations I have been given. So using the sin and cos identities, I can get rid of the x's. I know 2cos^2x = cos2x + 1 and that 2(cos(x))(sin(x)) = sin2x. I then added the a values back in so I have these two equations:
    x= a+a(cos(2x)) , y= a(sin(2x))
    From there, I fixed the x value to look like this:
    (x-a) = a(cos(2x)) , y = a(sin(2x))
    Then I remembered the circle equation of x^2 + y^2 = 1
    So I did this:
    (x-a)^2 + y^2 = a^2(cos^2(2x)) + a^2(sin^2(2x))
    I know I'm supposed to get (x-a)^2 + y^2 = a^2 , but I'm not sure what to do to the above equation to get this result. Any help would be greatly appreciated!
  2. jcsd
  3. Sep 17, 2008 #2
    Try factoring out a^2 from the right hand side.
  4. Sep 17, 2008 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What do you know about

    [tex] \sin^2 \theta + \cos^2 \theta [/tex] ?
  5. Sep 17, 2008 #4
    Oh, I get it. So I could write it like this:

    (x-a)^2 + y^2 = a^2 (cos^2(2x) + sin^2(2x))

    Then the identity takes care of the rest so I'm left with

    (x-a)^2 + y^2 = ^2 , right?
  6. Sep 17, 2008 #5
    I meant to write the above post as:

    (x-a)^2 + y^2 = a^2
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook