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            Let z = (-5sqrt(3)/2) + (5/2)i and w = 1 + sqrt(3)i            ...

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atrinity | (Level 1) Honors

Posted May 20, 2013 at 8:52 PM via web

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            Let z = (-5sqrt(3)/2) + (5/2)i and w = 1 + sqrt(3)i

 

            a. Convert z and w to polar form.

            b. Calculate zw.

            c. Calculate (z / w).

 

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mathsworkmusic | (Level 1) Educator

Posted May 20, 2013 at 9:40 PM (Answer #1)

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We have

`z = -(5sqrt(3))/2 + (5/2)i`,   `w = 1 + sqrt(3)i`

a) the polar form of a complex number consists of the argument and the modulus. Now,

`arg(z) = pi/2 + tan^(-1)(((5sqrt(3))/2)/(5/2)) = pi/2 + tan^(-1)(sqrt(3))`

`= pi/2 + pi/3 = (5pi)/6`

and  `|z| = sqrt(((5sqrt(3))/2)^2 + (5/2)^2) = sqrt(75/4 + 25/4) = sqrt(100/4) = 5`

Also,

`arg(w) = tan^(-1)(sqrt(3)) = pi/3`  and  `|w| = sqrt(1+sqrt(3)^2) = 2`

b) When complex numbers are multiplied together, their moduli multiply and their arguments add.

Since a complex number `u` can be written as

`u = |u|[cos(arg(u)) + isin(arg(u))]`

we have that

`zw = (5 times 2)(cos((5pi)/6 + pi/3) + isin((5pi)/6 + pi/3))`

`= 10cos((7pi)/6) + 10sin((7pi)/6)i` `= -5sqrt(3) -5i`

c) Simlarly

`z/w = (5/2)(cos((5pi)/6 - pi/3) + isin((5pi)/6 - pi/3))`

`= (5/2)cos(pi/2) + (5/2)sin(pi/2)i = 0 + (5/2)i`

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