How to use parametric equation/trigonometric identity to show an ellipse?I have the equation 16 x 2 + 25 y 2 = 400, and the parametric equation ( x , y ) = ( 5 cos ⁡ t , 4 sin ⁡ t )If I plug in the parametric equation into the first equation, I end up with the trigonometric identity cos 2 ⁡ t + sin 2 ⁡ t = 1. How does this identity show that my non-parametric equation, when graphed, will result in an ellipse?

Question

How to use parametric equation/trigonometric identity to show an ellipse?I have the equation   16      x    2    +  25      y    2    =  400, and the parametric equation   (  x  ,  y  )  =  (  5  cos  ⁡  t  ,  4  sin  ⁡  t  )If I plug in the parametric equation into the first equation, I end up with the trigonometric identity       cos    2    ⁡  t  +      sin    2    ⁡  t  =  1. How does this identity show that my non-parametric equation, when graphed, will result in an ellipse?

Jaydon Roth · Accepted Answer

The pair of parametric equations   x  =  5  cos  ⁡  t  ,  y  =  4  sin  ⁡  t in fact corresponds to the ellipse described by the Cartesian equation   16      x    2    +  25      y    2    =  400. Substituting the first equation into the second would naturally give you an identity (like you got). However, to properly prove this, it&#039;s better to manipulate one form into another.To prove this, manipulate the parametric equations:      x    2    =  25      cos    2    ⁡  t      1    25        x    2    =      cos    2    ⁡  tand similarly,      1    16        y    2    =      sin    2    ⁡  tAdd those up:      1    25        x    2    +      1    16        y    2    =      cos    2    ⁡  t  +      sin    2    ⁡  tAnd use the trigonometric identity       cos    2    ⁡  t  +      sin    2    ⁡  t  =  1 to get:      1    25        x    2    +      1    16        y    2    =  1Rearranging,  16      x    2    +  25      y    2    =  400, which is exactly the Cartesian equation you expect.You can see this is an ellipse by recognising the earlier derived canonical form:  (      x    a        )    2    +  (      y    b        )    2    =  1, where   a  =  5 and   b  =  4

Answered question

Answer & Explanation

New Questions in Trigonometry