Replace the Cartesian equations with equivalent polar equations. x 2 9 + y 2 4 = 1

Question

Replace the Cartesian equations with equivalent polar equations.            x              2              9    +            y              2              4    =  1

Kyle Delacruz · Accepted Answer

Since   x  =  r  cos  ⁡  θ and   y  =  r  sin  ⁡  θ , so an equivalent polar equation is                     r                  2                            cos                  2                    ⁡      θ        9    +                    r                  2                            sin                  2                    ⁡      θ        4    =  1. Aother is       r          2        =      36          4              cos                  2                    ⁡      θ      +      9              sin                  2                    ⁡      θ       if we multiply both sides of the equation by 36, factor out       r          2       and divide it by   4      cos          2        ⁡  θ  +  9      sin          2        ⁡  θ .Polar coordinates and Cartesian coordinates conversion:  r  =            x              2              +          y              2              θ  =      tan          −      1        ⁡  (      y    x    )  x  =  r  cos  ⁡  θ  y  =  r  sin  ⁡  θResult:      r          2        =      36          4      cos      ⁡      62      (      θ      )      +      9              sin                  2                    ⁡      (      θ      )

Raiden Barr · Accepted Answer

The goal of the exercise is to convert the given equation of cartesian coordinates            x              2              9    +            y              2              4    =  1into polar coordinates.Let&#039;s recall that the polar coordinates r and   θ are given as      r          2        =      x          2        +      y          2        ,  tan  ⁡  θ  =      y    x  ...(1)in terms of cartesian coordinates and the cartesian coordinates x and y are given as  x  =  r  cos  ⁡  θ  ,  y  =  r  sin  ⁡  θ...(2)To convert the given equation into cartesian coordinates let&#039;s substitute   x  =  r  cos  ⁡  θ and   y  =  r  sin  ⁡  θ in the given equation using the formula given in Eq. (2)            (      r      cos      ⁡      θ              )                  2                      9    +            (      r      sin      ⁡      θ              )                  2                      4    =  1                    r                  2                            cos                  2                    ⁡      θ        9    +                    r                  2                            sin                  2                    ⁡      θ        4    =  1which further can be simplified by taking the lcm on the left side expression            4              r                  2                            cos                  2                    ⁡      θ      +      9              r                  2                            sin                  2                    ⁡      θ        36    =  1or equally      r          2        (  4      cos          2        ⁡  θ  +  9  )      sin          2        ⁡  θ  =  36or equally      r          2        (  4      cos          2        ⁡  θ  +  4      sin          2        ⁡  θ  +  5      sin          2        ⁡  θ  )  =  36.Now we will use the trigonometric identity       cos          2        ⁡  θ  +      sin          2        ⁡  θ  =  1 to simplify the equation      r          2        (  4  ∗  1  +  5      sin          2        ⁡  θ  )  =  36      r          2        (  4  +  5      sin          2        ⁡  θ  )  =  36Result:      r          2        (  4  +  5      sin          2        ⁡  θ  )  =  36

Replace the Cartesian equations with equivalent polar equations. (x^(2))/(9)+(y^(2))/(4)=1

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