Rectangular vs Polar Coordinate Systems

Rectangular (Cartesian) Coordinates

• A point is written as (x,y)

• Position is determined by horizontal and vertical distances from the origin

Polar Coordinates

• A point is written as (r,θ)

• Position is determined by:

  • $r$: distance from the origin (radius)

  • θ: angle from the positive x-axis

Angles are typically measured in radians in AP Precalculus.

Understanding $r$ and θ

The Angle θ

• Measured counterclockwise from the positive x-axis

• Common reference angles:

  • $$\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2},\pi,2\pi$$

The Radius r

• Represents distance from the origin

• Can be positive or negative

• Negative r means the point lies in the opposite direction of θ

Example:

(2,π/3)and(−2,π/3)

These represent different points.

Converting Between Coordinate Systems

Polar to Rectangular

x=rcos⁡θ , y=rsin⁡θ

Example:
Convert (4,π/6):

$$x=4cos(\frac{\pi}{6})=2\sqrt{3} , y=4sin(\frac{\pi}{6})=2$$

Rectangular form:

$$(2\sqrt{3} , 2)$$

Rectangular to Polar

$$r=\sqrt{x^2+y^2} , \theta=tan^{-1}\left(\frac{y}{x}\right)$$

Quadrant awareness is critical when determining $\theta$.

Example:
Convert (−3,3):

$$r=\sqrt{(-3)^2+(3)^2}=\sqrt{18}=3\sqrt{2} , \theta=tan^{-1}\left(\frac{3}{-3}\right)=\frac{3\pi}{4}$$

Complex Numbers in Rectangular Form

A complex number in rectangular (standard) form is written as:

z=a+bi

where:

• $a$ is the real part

• $b$ is the imaginary part

• $$i=\sqrt{-1}$$​

Geometrically, the complex number corresponds to the point:

(a,b)

in the complex plane.

The Complex Plane and Polar Coordinates

Complex Plane

• Horizontal axis → real axis

• Vertical axis → imaginary axis

Each complex number corresponds to a point or vector from the origin.

Polar Coordinates

In polar form, a point is represented by: (r,θ)

where:

• r = distance from the origin (magnitude or modulus)

• θ = angle measured from the positive real axis

Polar Form of a Complex Number

A complex number in polar form is written as:

$$z=(\cos\theta+i\sin\theta)$$

Step-by-Step Conversion: Rectangular → Polar

Given:

z=a+bi

Step 1: Find the Magnitude r

The magnitude is the distance from the origin:

$$r=|z|=\sqrt{a^2+b^2}$$​

Step 2: Find the Argument $\theta$

$$\theta=\tan^{-1}\left(\frac{b}{a}\right)$$

⚠ Quadrant matters
You must adjust $\theta$ based on the location of the point (a,b).

Step 3: Write the Polar Form

$$z=(\cos\theta+i\sin\theta)$$

Examples

Example 1: First Quadrant

Convert:

z=3+3i

Magnitude:

$$r=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt{2}$$​

Angle:

$$\theta=\tan^{-1}\left(\frac{3}{3}\right)=\frac{\pi}{4}$$​

Polar Form:

$$z=3\sqrt{2}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})$$

Example 2: Second Quadrant

Convert:

z=−2+2i

Magnitude:

$$r=\sqrt{(-2)^2+2^2}\sqrt{8}=2\sqrt{2}$$​

Reference Angle:

$$\theta=\tan^{-1}\left(\frac{1}{-1}\right)=\tan^{-1}(-1)=\frac{3\pi}{4}$$​​

Polar Form:

$$z=2\sqrt{2}(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4})$$

Example 3: Fourth Quadrant

Convert:

$$z=1-\sqrt{3}i$$

Magnitude:

$$r=\sqrt{1^2+(\sqrt{3})^2}=2$$

Angle:

$$\theta=\tan^{-1}\left(\frac{1}{-\sqrt{3}}\right)=-\frac{\pi}{3}$$

Polar Form:

$$z=2(\cos(-\frac{\pi}{3})+i\sin(-\frac{\pi}{3}))$$

Special Cases

Purely Real Number

z=a

• r=∣a

• θ=0 or $\pi$

Purely Imaginary Number

z=bi

• r=∣b∣

• $$\theta=\frac{\pi}{2} or \frac{3\pi}{2}$$​​