Hello again! A few complex no. questions have got me stuck.
If z = 1 + i*root3
Find z bar to the power of -5.
If w = root2cis(pie/4)
Find w/z in polar and cartesian form and thus deduce an exact value for cos(pie/12).
z = 1 + root 3 i
z bar = 1 - root3 i
Converting this to cis form, z bar = 2 cis (-pi/3)
Using De Moivre's Theorem, (z bar)^5 = (2^5) cis 5*(-pi/3) = 32 cis (pi/3)
If w = root 2 cis (pi/4)
w/z in polar form would be (32/ root 2) cis (pi/3 - pi/4) = 16 root 2 cis (pi/12)
w/z in cartesian form would be:
w in cartesian form is root2 (cos pi/4 + i sin pi/4) = root 2 [(1/root 2) + i (1/root 2)] = 1 + i
w/z = (1+i)/(1 + root 3 i)
Realising the denominator would give you w/z = (1 + root 3)/4 + (1 - root 3)/4 * i
and thus deduce an exact value for cos(pi/12).
We know that 16 root 2 cis (pi/12) = 16 root 2 [cos (pi/12) + i sin (pi/12)] = (1 + root 3)/4 + (1 - root 3)/4 * i
So 16 root 2 * cos (pi/12) = (1 + root 3)/4
cos (pi/12) = (1 + root 3)/4 divided by 16 root 2...
Simplify and realise the answer.
Done!
I worked through the problem and still had trouble with the last part.
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)
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)
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, even after reading Alvin's comments.
Hello, classicstrings!
Both of you forgot the "minus" . . .
If .z .= .1 + i√3
Find z-bar to the power of -5.
We have: .z .= .2·cis(π/3) .2[cos(π/3) + i·sin(π/3)]
Then: .z-bar .= .2[cos(-π/3) + i·sin(-π/3)] .= .2[cos(π/3) - i·sin(π/3)]
Then: .(z-bar)^{-5} .= .(2^{-5})[cos(-5π/3) - i·sin(-5π/3)]
. . . . . . . . . . . . . . . = .(1/32)[(1/2) - i(√3/2)]
. . . . . . . . . . . . . . . = .(1/64)(1 - i√3)