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Points, Vectors, and Functions

Points, Vectors, and Functions







To translate an equation or inequality from rectangular to polar coordinates, x becomes rcosθ and y becomes rsinθ. We can also replace x2 + y2 with r2.

Translating from polar to rectangular coordinates isn't quite as straightforward as going the other way. We can replace r2 with x2 + y2, rcosθ with x, and rsinθ with y. However, sometimes we might need to do an extra step or two before we have any recognizable terms to replace.


x2 + y2 = 36

describes a pizza with a 6 inch radius, but isn't

r = 6

a much nicer way to describe it?

Going the other way,
the polar inequality

r ≤ 2/cosθ

is a mess. If we translate into rectangular coordinates, we find

rcosθ ≤ 2
x≤ &2



Example 1

Translate the equation x + y = 2x2 into polar coordinates.

Example 2

Translate the inequality x2 + y2 > 1 into polar coordinates.

Example 3

Translate the equation rcosθ + rsinθ = 2 into rectangular coordinates.

Example 4

Translate the inequality into rectangular coordinates, assuming .

Exercise 1

  • Translate each equation into polar coordinates.
  2. x + y2 = 0
  3. (x + y)2 = 4

Exercise 2

  • Translate each inequality into polar coordinates. %Graph the region being described by the inequality.
  1. y ≥ 5
  2. x + y≤ 3
  3. x2 + y2 < 100

Exercise 3

  • Translate each equation into rectangular coordinates.
  1. rcosθ + r2sin2θ = 4rsinθ
  2. (hint: multiply through by r)
  3. r = 4 (hint: square both sides.)

Exercise 4

  • Translate each inequality into rectangular coordinates. Assume r is non-negative.%Graph the region being described by the inequality.
  1. sin θ < 1/r
  2. rcos2θ + rsin2θ = 1
  3. 5r2cos2θ + 10rcosθ + 5 -4r2sin2θ + 16rsinθ-16 = 20
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