Finding z^4 in Polar & Cartesian Forms

[menco]

Homework Statement

Express z=-1+4i in polar for then find z^4 converting to Cartesian form

Homework Equations

r = sqrt(x^2+y^2)
theta = y/x
z= r cos (theta) + i r sin (theta)

The Attempt at a Solution

r= sqrt(-1^2+4^2)
= sqrt(17)

theta = tan a = 4/1
a = tan^-1 (4/1)
= 1.3258
pi-a = 1.8158 (to find the argument from the real axis)

in polar form = sqrt(17) cis 1.8158

Then to cartesian form

z^4 = r^4 cos 4(theta) + i r^4 sin 4(theta)
= (sqrt (17))^4 cos 4(1.8158) + i (sqrt(17))^4 sin 4(1.8158)
= 289 cos 7.2632 + i 289 sin 7.2632
z = 4sqrt(160.9760) + 4sqrt(240.0161) i
z = 3.5620 + 3.9360 i

Im not sure if this is correct but hopefully i am on the right track. I was not sure if I was converting to Cartesian form correctly or if i should introduce z^4 at the polar form so that z^4 = r^4 cis 4(theta)

 

[Mentallic]

z^4 = 289 cos 7.2632 + i 289 sin 7.2632
z = 4sqrt(160.9760) + 4sqrt(240.0161) i
z = 3.5620 + 3.9360 i

This is where you went wrong. You had found z⁴ correctly, but then in order to convert back to z
(which you shouldn't be doing anyway because the question asked you to convert z⁴ back into Cartesian form, not z)
you should have simply calculated $289\cdot \cos(7.2632)$ and $289\cdot \sin(7.2632)$

Now something else you should work on is avoiding the decimal approximations. If you calculate what you have, you won't get the exact answer. For example, $289\cdot \cos(7.2632)+i\cdot289\cdot\sin(7.2632)\approx 160.976+240.016i$ but the exact answer to the problem is $$\left(-1+4i\right)^4=161+240i$$

So let $\alpha = \tan^{-1}(4)$ therefore the argument is $\pi-\alpha$ and so we have

$$z=-1+4i=\sqrt{17}\left(\cos(\pi-\alpha)+i\sin(\pi-\alpha)\right)$$

And we can simplify these cos and sin expressions, and after that, we can then find z⁴. Now, can you simplify the expressions $\cos(\tan^{-1}\theta)$ and $\sin(\tan^{-1}\theta)$ ? And also, what about expressing $\cos(4x)$ and $\sin(4x)$ in terms of $\sin(x)$ and $\cos(x)$?

 

[grzz]

a = tan^-1 (4/1)
= 1.3258
pi-a = 1.8158 (to find the argument from the real axis)

I think that the argument required for z (which according to convention is measured from the real axis in an anticlockwise manner) is just a = tan^-1(4/1) and not (∏ - a).

 

[grzz]

I think that the argument required for z (which according to convention is measured from the real axis in an anticlockwise manner) is just a = tan^-1(4/1) and not (∏ - a).

I am so sorry!

I did not notice the -1 in z!

 

[Mentallic]

grzz, you can go back and edit your first post if you made a mistake somewhere. Better than making a second post