# Writing in polar form a complex number

1. Oct 22, 2012

### homevolend

1. The problem statement, all variables and given/known data

Write z = 1 + √3i in polar form

2. Relevant equations

z = r (cos$\varphi$ + sin$\varphi$i)

3. The attempt at a solution

Found the modulus by

|z| = √4 = 2

Now I am stuck on this part of finding the argument:

Tan-1 (√3)

now I am not sure how to go from that to the ans which is pi/3.

So would be:?

z= 2(cos(Tan-1 (√3)) + sin(Tan-1 (√3))i)

2. Oct 22, 2012

### phiiota

when i was in complex, the unit circle became my best friend. look at the unit circle, and think about the point that z would make on the unit circle.

3. Oct 22, 2012

### homevolend

√3 / 2 would be the y coordinate of pi/3.

Just not sure how that coordinate can make the argument pi/3?

4. Oct 22, 2012

### phiiota

well, the angle pi/3 gives you √3/2, correct? And you're multiplying on the outside by 2, right?

what's cos of the same angle?

you're using a different way to make your point. So you need angles to do it in the polar form.

If you memorize 3 or 4 pairs from the unit ciricle, I think, you can figure out most common forms of numbers they give you. Unless they're being sneaky, they usually give nice proportions of pi to make pretty numbers.

5. Oct 22, 2012

### phiiota

another way of thinking about this is, what is tangent? sin over cosine. so there needs to be some argument such that sin(x)/cos(x) =√3. So look at the unit circle. what arguments involve the root of 3? π/6 does, but tan of π/6=sin(π/6)/cos(π/6)=1/√3. What else does? π/3. now what do you get?