Writing in polar form a complex number

[homevolend]

Homework Statement

Write z = 1 + √3i in polar form

Homework Equations

z = r (cos\varphi + sin\varphii)

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)

 

[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.

 

[homevolend]

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

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

 

[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.

 

[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?