The complex number $-1 i/sqrt 3$ can be rewritten in the form $r (/cos/theta i/sin/theta),$ where $r / ge 0$ and $/theta/in[0,2/pi).$ What is $ (r,/theta)

Question:

The complex number - 1 + i√3 can be rewritten in the form r (cos (θ) + isin (θ) where θ is in the range [0, 2π]. What is r and θ?

Answer:

r = 2 and θ = 5π/3 radians

Polar form: 2 (cos (5π/3) + isin (5π/3))

Explanation:

The given complex number - 1 + i√3 is written in the rectangular form and we are asked to write it in the polar form so that we can get the values of r and θ.

The rectangular form is given by

x + iy

Where x is the real part and y is the imaginary part (i and j both are used denote the imaginary part of the complex number)

Comparing the given complex number to the standard rectangular form

x = - 1

y = √3

The polar form is given by

r (cos (θ) + isin (θ)

Where r is the magnitude of the complex number and θ is the angle

r can be found by using

r = √ (x²+y²)

r = √ ((-1) ² + (√3²)

r = √ (1+3)

r = √4

r = 2

θ can be found by using

θ = tan⁻¹ (y/x)

θ = tan⁻¹ (√3/-1)

θ = - 60°

Since x is negative an y is positive so the argument of complex number lies in the 2nd quadrant so we have to add π or 360°

θ = - 60° + 360° = 300°

We can convert the angle from the degrees into radians

θ = 300° (π/180°)

θ = 5π/3 radians

Therefore, r = 2 and θ = 5π/3 radians

Polar form: 2 (cos (5π/3) + isin (5π/3))