Replace the Cartesian equations with equivalent polar equations. x 2 + ( y − 2 ) 2 = 4

Question

Replace the Cartesian equations with equivalent polar equations.       x          2        +  (  y  −  2      )          2        =  4

Jean Deleon · Accepted Answer

Since   x  =  r  cos  ⁡  θ and   y  =  r  sin  ⁡  θ , so an equivalent polar equation is   (  r  cos  ⁡  θ      )          2        +  (  r  sin  ⁡  θ  −  2      )          2        =  4. This can be rewritten as       r          2            cos          2        ⁡  θ  +      r          2            sin          2        ⁡  θ  −  4  r  sin  ⁡  θ  +  4  =  4  →      r          2        (      cos          2        ⁡  θ  +      sin          2        ⁡  θ  )  −  4  r  sin  ⁡  θ  =  0  →      r          2        −  4  r  sin  ⁡  θ  =  0  →      r          2        =  4  r  sin  ⁡  θ . Divide both sides by r, we have another equivalent polar equation:   r  =  4  sin  ⁡  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  =  4  sin  ⁡  (  θ  )

Jacoby Erickson · Accepted Answer

The goal of the exercise is to convert the given equation of cartesian coordinates      x          2        +  (  y  −  2      )          2        =  4into 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        +  (  r  sin  ⁡  θ  −  2      )          2        =  4      r          2            cos          2        ⁡  θ  +  (      r          2            sin          2        ⁡  θ  +  4  −  4  r  sin  ⁡  θ  )  =  4      r          2        (      cos          2        ⁡  θ  +      sin          2        ⁡  θ  )  +  4  −  4  r  sin  ⁡  θ  =  4which further can be simplified by using the trigonometric identity       cos          2        ⁡  θ  +      sin          2        ⁡  θ  =  1      r          2        +  4  −  4  r  sin  ⁡  θ  =  4or equally      r          2        −  4  r  sin  ⁡  θ  =  0or equally  r  =  4  sin  ⁡  θ  ;Result:  r  =  4  sin  ⁡  θ

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

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