Lucas asked in Science & MathematicsMathematics · 4 weeks ago

# 9.Find the complex number with the given modulus and argument ,in the form x + iy.  a) Express z= -2-2(2¡)^1/2 [it's root] in polar form.?

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• 4 weeks ago

z = -2 - 2 * sqrt(2i)

z = -2 - 2 * sqrt(2) * sqrt(i)

Let's find the square root of i first

i^(1/2) =>

(0 + i)^(1/2) =>

(cos(pi/2 + 2pi * k) + i * sin(pi/2 + 2pi * k))^(1/2) =>

(cos((pi/2) * (1 + 4k)) + i * sin((pi/2) * (1 + 4k)))^(1/2) =>

cos((pi/2) * (1/2) * (1 + 4k)) + i * sin((1/2) * (pi/2) * (1 + 4k)) =>

cos((pi/4) * (1 + 4k)) + i * sin((pi/4) * (1 + 4k))

k is an integer.  Let's go with the principal value, so let k = 0

cos(pi/4) + i * sin(pi/4) =>

(sqrt(2)/2) + i * (sqrt(2)/2) =>

(sqrt(2)/2) * (1 + i)

z = -2 - 2 * sqrt(2) * sqrt(i)

z = -2 - 2 * sqrt(2) * (sqrt(2)/2) * (1 + i)

z = -2 - 2 * (1 + i)

z = -2 - 2 - 2i

z = -4 - 2i

Modulus

sqrt((-4)^2 + (-2)^2) = sqrt(16 + 4) = sqrt(20) = 2 * sqrt(5)

Argument

2 * sqrt(5) * cos(t) = -4

2 * sqrt(5) * sin(t) = -2

sin(t) = -2 / (2 * sqrt(5))

cos(t) = -4 / (2 * sqrt(5))

tan(t) = 1/2

t = arctan(1/2) + pi * k.  Again, k is an integer.

2 * sqrt(5) and the angle is arctan(1/2) + pi * k

• rotchm
Lv 7
4 weeks ago

Ambiguous. What does " z= -2-2(2¡)^1/2 [it's root] " mean?

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