Q. 23.7( 92 Votes )

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

z = – √3 + i

Answer :

As we know that the polar representation of a complex number z = x + iy is


z = r (cos θ + i sin θ) where is the modulus of the complex number and θ is the argument of the complex number, denoted by arg z.


So, now, let –√3 = r cos θ and 1 = r sin θ ……….(i)


Squaring both sides, we get


3 = r2 cos2 θ and 1 = r2 sin2 θ


Adding both the equations, we get


3 + 1 = r2 cos2 θ + r2 sin2 θ


4 = r2 (cos2 θ + sin2 θ)


4 = r2 or r2 = 4 [ sin2 θ + cos2 θ = 1]


r = 4


r = 2 (conventionally, r>0) ……….(ii)


Substituting r = 2 in (i), we get


–√3 = 2 cos θ and 1 = 2 sin θ




We know that the complex number –√3 + i lies in the second quadrant and the value of the argument lies between - π and π, i.e. - π < θ ≤ π.



……….(iii)


From (ii) and (iii), we have


r = 2 and

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Polar & Euler Form of Complex Numbers on Argand PlanePolar & Euler Form of Complex Numbers on Argand PlanePolar & Euler Form of Complex Numbers on Argand Plane32 mins
Practice session | Argument of complex numbersPractice session | Argument of complex numbersPractice session | Argument of complex numbers61 mins
Interactive Quiz Time - Polar & Euler Form of complex numberInteractive Quiz Time - Polar & Euler Form of complex numberInteractive Quiz Time - Polar & Euler Form of complex number58 mins
Modulus & Conjugate of Complex Number | Ready for a Quiz?Modulus & Conjugate of Complex Number | Ready for a Quiz?Modulus & Conjugate of Complex Number | Ready for a Quiz?48 mins
Questions on Modulus & Conjugate of Complex NumberQuestions on Modulus & Conjugate of Complex NumberQuestions on Modulus & Conjugate of Complex Number62 mins
Special Quiz on Argument of complex numbersSpecial Quiz on Argument of complex numbersSpecial Quiz on Argument of complex numbers56 mins
Polar & Euler Form of Complex Number on Argand PlanePolar & Euler Form of Complex Number on Argand PlanePolar & Euler Form of Complex Number on Argand Plane58 mins
Questions Based on Polar & Euler Form of Complex NumberQuestions Based on Polar & Euler Form of Complex NumberQuestions Based on Polar & Euler Form of Complex Number63 mins
Interactive Quiz on Quadratic EquationsInteractive Quiz on Quadratic EquationsInteractive Quiz on Quadratic Equations73 mins
Interactive Quiz on Quadratic Equations-02Interactive Quiz on Quadratic Equations-02Interactive Quiz on Quadratic Equations-0252 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses