complex numbers help?

Question

hi
i am quite stuck.
how do i give my answer in the form z = x + iy
when i know the modulus |z| = 5
and the argument = pi/8

Answer 1

you need half angle formula...

z = 5 (cis (pi/8))
= 5 (cos(pi/8) + i sin(pi/8))

cos(pi/8) = [ (1 + cos(pi/4))/2 ]^1/2
= (1/2)*(2 + sqrt(2))^1/2

sin(pi/8) = [ (1 - cos(pi/4))/2 ]^1/2
= (1/2)*(2 - sqrt(2))^1/2

z = [(5/2)*(2 + sqrt(2))^1/2] + i [(5/2)*(2 - sqrt(2))^1/2]



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Answer 2

Greetings,

|z| = 5

recall that z = sqrt (x^2 + y^2)

z = |z|(x /|z| + iy/|z|)

z = 5( x/sqrt(x^2 + y^2) + iy/sqrt (x^2 + y^2))

These are the sin and cos values

z = 5 (cos (pi/8) + isin(pi/8))

z = 5 cos(pi/8) + 5sin(pi/8) i

Regards

Answer 3

The previous answer should be

5 cos(pi/8) + [5 sin(pi/8)]i

Answer 4

|z| = 5
x^2 + y^2 = 5

argument = pi/8
tan(pi/8) = y/x
y = x tan(pi/8)

x^2 + [x tan(pi/8)]^2 = 5
x^2 = 4.2677669529663688110021109052621
x = 2.07 (3 sig fig)

y
= x tan(pi/8)
= 0.85570616863128384777481807182468
= 0.856 (3 sig fig)

Answer 5

x = 5 cos π/8 = 4.62
y = 5 sin π/8 = 1.91

z = 4.62 + i 1.91