Polar equations and their graphs

About this lesson

Precalculus › Parametric equations and polar coordinates

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Polar equations and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Which points satisfy the polar equation r=3r = 3?

  1. All points at distance 3 from the origin
  2. The points on one line through the origin

Key idea

A polar equation links rr and θ\theta. To recognize it, convert to xx and yy.

Use x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta and r2=x2+y2r^2 = x^2 + y^2. Classic graphs: r=ar = a is a circle around the origin; θ=α\theta = \alpha is a line through the origin with slope tan⁡α\tan\alpha; r=2asin⁡θr = 2a\sin\theta is a circle of radius ∣a∣|a| centered at (0,a)(0, a).

Worked example

Identify r=6cos⁡θr = 6\cos\theta.

r2=6rcos⁡θr^2 = 6r\cos\theta
Multiply both sides by r.
x2+y2=6xx^2 + y^2 = 6x
r² = x² + y², r cos θ = x.
(x−3)2+y2=9(x - 3)^2 + y^2 = 9: a circle, center (3,0)(3, 0), radius 3
Complete the square in x.

Your turn, guided

Your turn: r=8sin⁡θr = 8\sin\theta.

x2+y2=8yx^2 + y^2 = 8y
Multiply by r, convert.
x2+(y−4)2=16x^2 + (y - 4)^2 = 16: radius 44
(y − 4)² − 16: type the radius.

Practice ★

  1. What is the graph of the polar equation r=5r = 5?
    1. A circle of radius 5 centered at the origin
    2. A line through the origin
    3. The vertical line x=5x = 5
  2. What is the graph of the polar equation r=7r = 7?
    1. A circle of radius 7 centered at the origin
    2. A line through the origin
    3. The vertical line x=7x = 7

Watch out!

Practice ★★

  1. The polar equation r=12sin⁡θr = 12\sin\theta is a circle. In rectangular coordinates, it is x2+y2=12yx^2 + y^2 = 12y. What is the yy-coordinate of its center?
  2. The polar equation r=6sin⁡θr = 6\sin\theta is a circle. In rectangular coordinates, it is x2+y2=6yx^2 + y^2 = 6y. What is the yy-coordinate of its center?
  3. The polar equation r=8sin⁡θr = 8\sin\theta is a circle. In rectangular coordinates, it is x2+y2=8yx^2 + y^2 = 8y. What is the yy-coordinate of its center?
  1. The polar equation θ=π6\theta =\frac{\pi}{6} is a line through the origin. What is its slope?
  2. The polar equation θ=5π6\theta =\frac{5 \pi}{6} is a line through the origin. What is its slope?

Exit ticket

Three questions to check you have got it.

  1. The polar equation r=12sin⁡θr = 12\sin\theta is a circle. In rectangular coordinates, it is x2+y2=12yx^2 + y^2 = 12y. What is the yy-coordinate of its center?
  2. The polar equation r=6sin⁡θr = 6\sin\theta is a circle. In rectangular coordinates, it is x2+y2=6yx^2 + y^2 = 6y. What is the yy-coordinate of its center?
  3. The polar equation r=8sin⁡θr = 8\sin\theta is a circle. In rectangular coordinates, it is x2+y2=8yx^2 + y^2 = 8y. What is the yy-coordinate of its center?

Summary card

Polar equations

  • Convert with x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta, r2=x2+y2r^2 = x^2 + y^2.
  • r=ar = a: circle; θ=α\theta = \alpha: line; r=2asin⁡θr = 2a\sin\theta: circle of radius ∣a∣|a| through the origin.

Watch out: in r=2asin⁡θr = 2a\sin\theta, 2a2a is the diameter.

End of the lesson

Next lesson: Complex numbers in polar form Back to the chapter

Answers: Polar equations and their graphs

A

  1. A circle of radius 5 centered at the origin
  2. A circle of radius 7 centered at the origin

B

  1. 66
  2. 33
  3. 44

C

  1. 33\frac{\sqrt{3}}{3}
  2. −33- \frac{\sqrt{3}}{3}

D

  1. 66
  2. 33
  3. 44

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