Aim for FivePrecalculus

Unit 3 · Topic 3.13

3.13 Trigonometry and Polar Coordinates

Polar coordinates locate a point by its distance from the origin and an angle, written (r, θ), instead of by x and y. You'll convert between polar and rectangular coordinates and write complex numbers in polar form.

Key terms

  • polar coordinates
  • polar axis
  • rectangular coordinates
  • complex plane
  • polar form of a complex number

The polar grid

Picture a target: circles centered at the origin, crossed by lines through the origin. That's the polar grid. The origin is called the pole, and the positive x-axis is called the polar axis.

A point is written (r, θ). The angle θ is in standard position, and r is the signed distance from the origin along the line at that angle.

If r > 0, go r units along the terminal ray of θ. If r < 0, go |r| units in the opposite direction, along the ray for θ + π. So (−2, π/3) is the same point as (2, 4π/3).

Every point has infinitely many polar names. (3, π/4), (3, 9π/4), (3, −7π/4) and (−3, 5π/4) all name the same point.

Polar coordinates are natural for anything that turns or spreads out from a center, like a radar screen, a lawn sprinkler, or a point on a spinning wheel. A circle centered at the origin, which needs x² + y² = 9 in rectangular form, is just r = 3 in polar form.

Polar to rectangular

From (r, θ), use x = r cos θ and y = r sin θ. These come from the definitions of sine and cosine on a circle of radius r, and they work for negative r too.

Rectangular to polar

From (x, y), the distance is r = √(x² + y²).

For the angle, tan θ = y/x, but arctan only returns angles between −π/2 and π/2, which point to the right of the y-axis. So:

  • If x > 0: θ = arctan(y/x).
  • If x < 0: θ = arctan(y/x) + π.
  • If x = 0: θ = π/2 for points above the origin and 3π/2 (or −π/2) for points below.

Complex numbers in polar form

The complex number a + bi can be plotted as the point (a, b) in the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.

That point also has polar coordinates (r, θ), with r = √(a² + b²). Since a = r cos θ and b = r sin θ, the number can be written in polar form as (r cos θ) + (r sin θ)i.

The value r is the distance from 0 to the number in the complex plane, also called its modulus.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1

    Polar to rectangular

    Convert (4, 5π/6) and (−2, π/3) to rectangular coordinates.

    Show the solution
    1. Step 1: (4, 5π/6): x = 4 cos(5π/6) = 4(−√3/2) = −2√3, and y = 4 sin(5π/6) = 4(1/2) = 2.
    2. Step 2: (−2, π/3): x = −2 cos(π/3) = −2(1/2) = −1, and y = −2 sin(π/3) = −2(√3/2) = −√3.
    3. Step 3: The second point is in quadrant III, even though π/3 points into quadrant I, because r is negative.

    Answer: (−2√3, 2) and (−1, −√3).

  2. Example 2

    Rectangular to polar

    Convert (−3, 3) to polar coordinates with r > 0 and 0 ≤ θ < 2π.

    Show the solution
    1. Step 1: r = √(9 + 9) = √18 = 3√2.
    2. Step 2: x < 0, so θ = arctan(3/(−3)) + π = arctan(−1) + π = −π/4 + π = 3π/4.
    3. Step 3: Check: 3π/4 points into quadrant II, where (−3, 3) is.

    Answer: (3√2, 3π/4).

  3. Example 3

    Trap: a complex number in the wrong quadrant

    Write −1 − √3 i in polar form.

    Show the solution
    1. Step 1: The point is (−1, −√3), in quadrant III. r = √(1 + 3) = 2.
    2. Step 2: arctan((−√3)/(−1)) = arctan(√3) = π/3. That angle points into quadrant I, the opposite direction. Since x < 0, add π: θ = 4π/3.
    3. Step 3: Polar form: 2 cos(4π/3) + (2 sin(4π/3))i. Check: 2 cos(4π/3) = −1 and 2 sin(4π/3) = −√3.

    Answer: −1 − √3 i = 2 cos(4π/3) + (2 sin(4π/3))i.

Common mistakes

  • Using θ = arctan(y/x) for a point with x < 0. Add π.
  • Plotting (−r, θ) in the direction of θ. A negative r points the opposite way.
  • Mixing up the conversion formulas: x goes with cosine, y with sine.

On the exam

  • Expect conversions in both directions, often with exact values, and questions about which polar pairs name the same point.
  • Complex numbers in polar form may appear as a point to convert, so practice reading a + bi as (a, b).

Connected topics

Videos

  • AP Precalculus – 3.13 Trigonometric and Polar Coordinates

    The AlgebrosWatch on YouTube (opens in a new tab)

  • Converting polar and rectangular coordinates | Precalculus | Khan Academy

    Khan AcademyWatch on YouTube (opens in a new tab)

  • Trigonometric Form of a Complex Number

    Mario's Math TutoringWatch on YouTube (opens in a new tab)

  • Intro to Polar Function in Under 3 mins (AP Precalculus Unit 3 Topic 3.13)

    Maximum InsightWatch on YouTube (opens in a new tab)

  • 3.13-A Trig and Polar Coordinates Video

    Flamingo Math by Jean AdamsWatch on YouTube (opens in a new tab)

  • Polar Coordinates Basic Introduction, Conversion to Rectangular, How to Plot Points, Negative R Valu

    The Organic Chemistry TutorWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 3.13 Trigonometry and Polar Coordinates. Pick an answer to see if you got it, and why.

Question 1 of 4

A point has polar coordinates (r, θ) = (−4, π/3). What are its rectangular coordinates?

Question 2 of 4

Which of the following is the complex number −1 − √3 i written in polar form?

Question 3 of 4Calculator allowed

A point has polar coordinates (5, 2.2), where the angle is in radians. Which of the following is closest to the rectangular coordinates of the point?

Question 4 of 4

Which of the following gives polar coordinates (r, θ), with r > 0 and 0 ≤ θ < 2π, for the point with rectangular coordinates (−3, −3)?

0 of 4 answered