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4 April, 16:58

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)

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  1. 4 April, 20:30
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    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))
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