Changing complex numbers in form a+bi

[ME_student]

Homework Statement

Here is the problem: ($\sqrt{}6$(cos(3pie/16)+i sin(3pie/16)))^4

Homework Equations

After 360*=2pier Do you add 2pier to the next degree which would be 30*=pie/6?

The Attempt at a Solution

 

[daveb]

In situations such as these, it is usually easiest to convert to the exponential form and solve from there, then convert back to a+bi form.

 

[vela]

After 360*=2pier Do you add 2pier to the next degree which would be 30*=pie/6?

What do you mean by this? By "2pier" do you mean $2\pi$ radians or $2\pi r$, the circumference of a circle? What is "the next degree"? After 360 degrees, the next degree is 361 degrees, no?

 

[Muphrid]

The greek letter $\pi$ is just spelled as "pi". It is not the same as the dish.

 

[ME_student]

What do you mean by this? By "2pier" do you mean $2\pi$ radians or $2\pi r$, the circumference of a circle? What is "the next degree"? After 360 degrees, the next degree is 361 degrees, no?

Sorry, I couldn't find the 2 "pie" r.

No worry guys I figured it out. Wish I could post pictures... I would show you how I solved it.

EDIT: Sweet I can post pics now!

 

[ME_student]

Here is the problem

 

[Muphrid]

Per daveb, I too think it would be much easier to convert to polar form first and then convert back. The reason is that the exponential lends itself to being raised to a power much more easily than the rectangular form.

What you get is

$$(\sqrt{6} e^{3\pi i/16})^4 = 36 e^{3\pi i/4}$$

This is what you got, after all, but you can do it in one line as opposed to resorting to a bunch of trig identities to square the rectangular form.

 

[ME_student]

Apologies for the quality of the pictures...

 

[AmritpalS]

Look up de Moivre's Theorem. It'll narrow down the tedious computation with multplying it out.

 

[BloodyFrozen]

$$(cos(x)+i~sin(x))^{n} = cos(nx)+i~sin(nx)$$

 

[ME_student]

Whoa... I was a member while taking trig, crazy!

 

[2milehi]

So much easier when using Euler's formula. I can do it in my head rather quickly.

 

[ME_student]

So much easier when using Euler's formula. I can do it in my head rather quickly.

Good for you.

 

[2milehi]

Just sayin' - if you recognize this

then the problem become easy to do. There are plenty of ways to skin the cat with this problem.