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Convert Equations Between Polar and Rectangular Forms quiz

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  • What substitutions are used to convert x and y in rectangular equations to polar form?
    x is replaced with r*cos(theta) and y is replaced with r*sin(theta).
  • How do you convert y = 5 to polar form?
    Replace y with r*sin(theta), solve for r, and get r = 5*csc(theta).
  • What is the polar form of y = x + 1?
    The polar form is r = 1 / (sin(theta) - cos(theta)).
  • How do you convert x^2 + y^2 = 25 to polar form?
    Replace x^2 + y^2 with r^2, then solve to get r = 5.
  • What is the rectangular equivalent of r = 4?
    Square both sides to get r^2 = 16, then replace r^2 with x^2 + y^2 to get x^2 + y^2 = 16.
  • How do you convert r = sec(theta) to rectangular form?
    Rewrite sec(theta) as 1/cos(theta), multiply both sides by cos(theta), and replace r*cos(theta) with x to get x = 1.
  • What is the rectangular form of r = 6*sin(theta)?
    Multiply both sides by r to get r^2 = 6r*sin(theta), then replace r^2 with x^2 + y^2 and r*sin(theta) with y, resulting in x^2 + y^2 = 6y.
  • What strategy is used to rewrite x^2 + y^2 = 6y into a standard form?
    Complete the square for the y terms to get x^2 + (y - 3)^2 = 9.
  • What is the standard form of a circle in rectangular coordinates?
    The standard form is (x - h)^2 + (y - k)^2 = r^2.
  • What is the polar form of a circle centered at the origin with radius 5?
    The polar form is r = 5.
  • How do you convert r*cos(theta) = 1 to rectangular form?
    Replace r*cos(theta) with x to get x = 1.
  • What manipulation is often used when a polar equation contains a fraction?
    Multiply both sides by the denominator to eliminate the fraction.
  • What is the rectangular form of r = 5*csc(theta)?
    Replace r with y/sin(theta), solve for y, and get y = 5.
  • What is the rectangular form of r = 1/(sin(theta) - cos(theta))?
    Replace r*sin(theta) with y and r*cos(theta) with x, rearrange, and get y - x = 1.
  • What is the purpose of completing the square when converting equations?
    Completing the square helps rewrite equations into standard forms, such as circles, for easier identification and graphing.