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## Homework Statement

If z = 1 + i*root3

i) Find the modulus and argument of z

ii) Express z^5 in Cartesian form a + ib where a and b are real

iii) Find z*zbar

iv) hence, find zbar to the power of negative 5

If w = root2 cis (π/4)

i) Find w/z in polar form.

ii) Express z and w in cartesian form and hence find w/z in cartesian form.

iii) Use answers from i) and ii) to deduce exact value for cos(π/12)

## The Attempt at a Solution

i) Modulus is 2, Argument is π/3

ii) Using De Moivre's Theorem, 2^5cis(5*(π/3))

= 32cis(-pie/3)

Then changing to Cartesian form - 32 (cos(-π/3) + isin(-π/3))

= 32(0.5 + (root3/2)i)

= 16 + 16i*root3

iii) z = 1 + i*root3, zbar = 1 - i*root3

z*zbar = 4

iv) Now this part I don't understand the "hence", as in how am I meant to use previous results to get this answer?

I try - zbar = 2cis(-π/3) -> (Is it true that the "bar" of any complex number is just the same modulus and negative angle?)

Then zbar^-5 = Using DM Theorem, (1/32)cis(5π/3)

= (1/32)cis(-π/3)

Then convert to cartesian - (1/32)(cos(-π/3) + isin(-π/3))

= (1/32)(0.5 + iroot3/2)

= 1/64 + iroot3/64

If w = root2 cis (π/4)

i) Find w/z in polar form.

I found it in cartesian - (1+i)/(1 + root3 i) then realising, gives (1 + root3)/4 + (1-root3)/4 * i

But how do I change to Polar form? I am also stuck on how to get cos(pie/12) as exact value. Thanks.