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erpoi
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I was wondering if someone can check my solutions and perhaps give me a faster more logical way of working through this question. Thanks.
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)
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.
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.