Show that x ( t ) = 2 r cos 2 ⁡ ( t ) , y ( t ) = 2 r sin ⁡ t cos ⁡ t is a regular parametrization of the real circle of radius r, centre (r, 0).

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Show that   x  (  t  )  =  2  r      cos    2    ⁡  (  t  ) ,   y  (  t  )  =  2  r  sin  ⁡  t  cos  ⁡  t is a regular parametrization of the real circle of radius r, centre (r, 0).

zlepljalz2 · Accepted Answer

Step 1Because of the identities       cos    2    ⁡  u  =  (  1  +  cos  ⁡  2  u  )      /    2 and   sin  ⁡  2  u  =  2  sin  ⁡  u  cos  ⁡  u sinucosu, your parametric equations are equivalent to                    x        (        t        )                            =        r        +        r        cos        ⁡        2        t                            y        (        t        )                            =        r        sin        ⁡        2        t            x(t)y(t)=r+rcos2t=rsin2tAnd now you can see that this describes the circle centred (r, 0) with radius r.Yes, the parameterisation is twice as fast as you&#039;d expect (with 2t in the expressions), but this is okay as parametrisations are not unique.

Show that x ( t ) = 2 r cos 2 </msup> ⁡

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