Convert the polar point (2, π) to rectangular coordinates. [1 point]

College Board AP 2023 (style)-style practice: Section I, Part A (multiple choice, no calculator). The polar point (4, (π)/(3)) has which rectangular coordinates? (A) (2, 2sqrt(3)) (B) (2sqrt(3), 2) (C) (4, (π)/(3)) (D) (2sqrt(3), 4)

The correct answer is (A), (2, 2sqrt(3)). Convert with x = rcosθ and y = rsinθ. Here r = 4, θ = (π)/(3), so x = 4cos(π)/(3) = 4 · (1)/(2) = 2 and y = 4sin(π)/(3) = 4 · (sqrt(3))/(2) = 2sqrt(3). The rectangular point is (2, 2sqrt(3)).

United StatesPrecalculusSyllabus dot point

What are polar coordinates, and how do you convert between polar and rectangular form?

Topic 3.13 Trigonometry and Polar Coordinates: locate points using polar coordinates and convert between polar and rectangular coordinates.

A focused answer to AP Precalculus Topic 3.13, covering the polar coordinate system, how a point is named by radius and angle, the conversion formulas between polar and rectangular coordinates, and why polar names are not unique.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

Polar coordinates name a point as $(r, \theta)$ , where $r$ is the directed distance from the origin (the pole) and $\theta$ is the angle of the ray from the positive $x$ -axis (the polar axis). To convert polar to rectangular, use $x = r\cos\theta$ and $y = r\sin\theta$ , which come straight from the unit-circle definitions scaled by $r$ . To convert rectangular to polar, use $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\frac{y}{x}$ , adjusting the angle for the correct quadrant. Unlike rectangular coordinates, a polar name is not unique: adding $2\pi$ to the angle, or negating $r$ while rotating the angle by $\pi$ , names the same point. This many-to-one feature is the key difference from the Cartesian system and matters when reading polar graphs.

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What this topic is asking
The polar coordinate system
Converting polar to rectangular
Converting rectangular to polar
Why polar names are not unique
Try this

What this topic is asking

The College Board (Topic 3.13) wants you to use polar coordinates, which name a point by its distance from the origin and the angle of its direction, and to convert between polar and rectangular (Cartesian) coordinates. The conversion runs through the trigonometric definitions of sine and cosine, tying this topic back to the unit circle.

The polar coordinate system

So polar coordinates describe location by "how far and in which direction", whereas rectangular coordinates use "how far right and how far up".

Converting polar to rectangular

This direction is always straightforward: plug $r$ and $\theta$ into the two formulas.

Converting rectangular to polar

The quadrant adjustment is the step most often missed: arctangent cannot tell Quadrant II from Quadrant IV, so you must check the signs of $x$ and $y$ and place $\theta$ accordingly.

Converting a rectangular point to polar

Convert $(-2, -2\sqrt{3})$ to polar form with $r > 0$ and $0 \le \theta < 2\pi$ .

step 1 Find the radius

$r = \sqrt{(-2)^2 + (-2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4$ .

step 2 Find the reference angle

$\tan\theta = \frac{-2\sqrt{3}}{-2} = \sqrt{3}$ , so the reference angle is $\frac{\pi}{3}$ .

step 3 Place the angle in the correct quadrant

Both coordinates are negative, so the point is in Quadrant III. The Quadrant III angle with reference $\frac{\pi}{3}$ is $\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$ .

step 4 State the polar point

$\left(4, \frac{4\pi}{3}\right)$ . Check: $x = 4\cos\frac{4\pi}{3} = 4\left(-\frac{1}{2}\right) = -2$ and $y = 4\sin\frac{4\pi}{3} = 4\left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}$ , matching the original point.

Why polar names are not unique

A point worth stating once is that a single point has infinitely many polar names. Adding any multiple of $2\pi$ to the angle lands on the same ray, so $(r, \theta)$ and $(r, \theta + 2\pi k)$ coincide. A negative radius reverses the direction, so $(-r, \theta)$ equals $(r, \theta + \pi)$ . The origin itself is $(0, \theta)$ for every $\theta$ . This non-uniqueness is harmless when plotting a single point but becomes important when graphing a polar function (Topic 3.14), where the same point can be traced more than once. Always state any required restrictions on $r$ and $\theta$ when a unique answer is wanted.

Try this

Q1. Convert the polar point $\left(2, \pi\right)$ to rectangular coordinates. [1 point]

Cue. $x = 2\cos\pi = -2$ , $y = 2\sin\pi = 0$ , giving $(-2, 0)$ .

Q2. What is the radius $r$ for the rectangular point $(3, 4)$ ? [1 point]

Cue. $r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part A (multiple choice, no calculator). The polar point

\left(4, \frac{\pi}{3}\right)

has which rectangular coordinates? (A)

(2, 2\sqrt{3})

(B)

(2\sqrt{3}, 2)

(C)

(4, \frac{\pi}{3})

(D)

(2\sqrt{3}, 4)

Show worked answer →

The correct answer is (A), $(2, 2\sqrt{3})$ .

Convert with $x = r\cos\theta$ and $y = r\sin\theta$ . Here $r = 4$ , $\theta = \frac{\pi}{3}$ , so $x = 4\cos\frac{\pi}{3} = 4 \cdot \frac{1}{2} = 2$ and $y = 4\sin\frac{\pi}{3} = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}$ . The rectangular point is $(2, 2\sqrt{3})$ .

AP 2024 (style)3 marksSection II (free response, no calculator). A point has rectangular coordinates

(-3, 3)

. (a) Find a polar representation

(r, \theta)

with

r > 0

and

0 \le \theta < 2\pi

. (b) Give a second polar representation of the same point.

Show worked answer →

A 3-point question on polar conversion.

(a) Primary representation (2 points): $r = \sqrt{(-3)^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$ . The point is in Quadrant II, and $\tan\theta = \frac{3}{-3} = -1$ with the angle in Quadrant II gives $\theta = \frac{3\pi}{4}$ . So $\left(3\sqrt{2}, \frac{3\pi}{4}\right)$ .
(b) Second representation (1 point): adding $2\pi$ to the angle gives the same point, $\left(3\sqrt{2}, \frac{3\pi}{4} + 2\pi\right) = \left(3\sqrt{2}, \frac{11\pi}{4}\right)$ . (A negative-radius form $\left(-3\sqrt{2}, \frac{3\pi}{4} - \pi\right) = \left(-3\sqrt{2}, -\frac{\pi}{4}\right)$ also names the same point.)

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Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)