Question

Express z=−10\sqrt{3} −10i in polar form.

Express your answer in exact terms, using radians, where your angle is between 0 and 2π radians, inclusive.

Answer 1

z = [20/sqrt(3][cos(-2π/3) + isin(-2π/3)]

Explanation:

Polar form is

r(cos(theta) + isin(theta))

r = sqrt[(-10/sqrt3)² + (10)²]

r = sqrt(400/3)

r = 20/sqrt(3)

tan^-1[10 ÷ (10/sqrt(3))] = π/3

Theta: -π + π/3 = -2π/3

z = [20/sqrt(3][cos(-2π/3) + isin(-2π/3)]

Answer 2

11pi/6

Explanation:

10sqrt3 - 10i = a(cos(b)+sin(b)i)

square both terms to find a

300 + 100 = (a^2)

400 = a^2

20 = a

20(cos(b) + sin(b)i) = 10sqrt3 - 10i

cos(b) = (sqrt3)/2; sin(b) = -1/2

THE REFERENCE ANGLE WOULD BE pi/6

and...

THE ONLY QUADRANT IN WHICH COS IS POSITIVE AND SIN IS NEGATIVE IS THE Q4

That makes b = 2pi - (pi/6) = 11pi/6