Converting Rectangular form to Polar form

Sal's video is teaching how to convert from rectangular form (a+bi) to polar form, which is written as:

z=r(cos⁡θ+isin⁡θ)z=r(cosθ+isinθ)

where:

• rr = distance from the origin (the modulus)

θθ = angle from the positive x-axis (the argument).

Khan Academy+1

The idea visually

Think of 3 + 4i as a point on a graph.

Instead of describing the point as "go 3 right and 4 up," polar form says:

"Go 5 units from the origin at an angle of about 53.1°."

Same point, different description.

Step 1: Find the modulus rr

Use the Pythagorean theorem.

r=a2+b2r=a2+b2​

For 3+4i:

r=32+42=9+16=5r=32+42​=9+16​=5

Nice. It's the classic 3-4-5 triangle.

Step 2: Find the angle θθ

Use tangent.

tan⁡θ=batanθ=ab​

So

tan⁡θ=43tanθ=34​

Take the inverse tangent:

θ≈53.1∘θ≈53.1∘

Important: This only gives the reference angle. You still have to check which quadrant the number is actually in.

Khan Academy+1

Step 3: Write the polar form

Put the two pieces together.

3+4i=5(cos⁡53.1∘+isin⁡53.1∘)3+4i=5(cos53.1∘+isin53.1∘)

Done.

The sneaky quadrant trick

This is where people get caught.

Take -3+4i.

The calculator says

arctan⁡(4−3)=−53.1∘arctan(−34​)=−53.1∘

But that's wrong for this point!

Why?

Because -3+4i is in Quadrant II.

So the actual angle is

180∘−53.1∘=126.9∘180∘−53.1∘=126.9∘

Always ask yourself:

Quick cheat sheet

Tiny challenge (your turn 😈)

Convert −4−4i−4−4i into polar form.

Try finding:

1. rr

2. the reference angle

3. the correct quadrant

4. the final polar form

I won't spoil it immediately—I have a feeling you'll probably get this one.