Trigonometry and Polar Coordinates

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

Polar coordinates locate a point by distance $r$ from the origin and angle $\theta$ from the positive x-axis. Conversion to rectangular is straightforward: $x = r \cos\theta$ , $y = r \sin\theta$ . Converting back is trickier — $r = \sqrt{x^2 + y^2}$ , but $\theta$ from $\arctan(y/x)$ must be adjusted for the actual quadrant of $(x, y)$ .

Conversion

Polar → Rectangular

x = r \cos\theta, \;\; y = r \sin\theta

Rectangular → Polar

r = \sqrt{x^2 + y^2}, \;\; \theta = \arctan\!\left(\tfrac{y}{x}\right) \text{ (quadrant-adjusted)}

Caution:

\arctan(y/x)

only gives the principal value. For points in Q2 or Q3, add

\pi

to get the correct

\theta

Type 1

Convert (x, y) to (r, θ)

Compute $r$ from the Pythagorean formula, then determine $\theta$ using arctan and the quadrant.

Example 1

Convert the rectangular point

(-1, \sqrt{3})

to polar with

r \geq 0

and

\theta \in [0, 2\pi)

. (A)

(2, \pi/3)

(B)

(2, 2\pi/3)

(C)

(2, 4\pi/3)

(D)

(2, 5\pi/3)

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Type 2

Convert (r, θ) to (x, y)

Apply $x = r\cos\theta$ , $y = r\sin\theta$ . Use exact values at special angles.

Example 2

Convert

(2, \pi/6)

from polar to rectangular. (A)

(1, \sqrt 3)

(B)

(\sqrt 3, 1)

(C)

(2, 1)

(D)

(\sqrt 3, 2)

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Type 3

Non-uniqueness of polar coordinates

$(r, \theta)$ and $(r, \theta + 2\pi)$ are the same point. So is $(-r, \theta + \pi)$ .

Example 3

Which of the following represent the same point as

(3, \pi/4)

? (A)

(3, 5\pi/4)

(B)

(-3, 5\pi/4)

(C)

(3, 9\pi/4)

only (D)

(-3, 5\pi/4)

and

(3, 9\pi/4)

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