How do you find the Cartesian equation of the curve with parametric equations x = 2 cos ( 3 t ) and y = 2 sin ( 3 t ) , and determine the domain and range of the corresponding relation?

Question

How do you find the Cartesian equation of the curve with parametric equations       x    =    2          cos              (        3        t        )             and       y    =    2          sin              (        3        t        )             , and determine the domain and range of the corresponding relation?

Kiana Cantu · Accepted Answer

Step 1The domain and range are easily obtained before converting to the Cartesian Equation.Please observe that the range cosine function,       -    1    ≤          cos              (        u        )              ≤    1   , causes the domain for x to be:      -    2    ≤    x    ≤    2  The range for y is limited in the same way but with the sine function:      -    2    ≤    y    ≤    2  Use the equation for x to find an equation for             sin              (        3        t        )                  x    =    2          cos              (        3        t        )            Use the identity             cos              (        u        )              =    ±                  1        -                              sin            2                                (            u            )                               :Because we are going to square everything, we shall only use the positive value:      x    =    2                  1        -                              sin            2                                (            3            t            )                                                  x        2            4        =    1    -                  sin        2                    (        3        t        )                                sin        2                    (        3        t        )              =    1    -                  x        2            4                  sin              (        3        t        )              =    ±                  1        -                              x            2                    4                          y    =          {                                    2                                          1                -                                                      x                    2                                    4                                                                                          -            2                                          1                -                                                      x                    2                                    4                                                                          ;    -    2    ≤    x    ≤    2  This should look like a circle to you, therefore, a better Cartesian form is found by squaring y:            y      2        =    4          (      1      -                        x          2                4            )                  x      2        +          y      2        =          2      2

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