Represent the complex number z=3+iz=3+i in the polar form.


Answer:

2(cosπ6+i sinπ6)2(cosπ6+i sinπ6)

Step by Step Explanation:
  1. We have, z=3+i The standard polar form of a complex number is r(cosθ+i sinθ)
    Θ y y' x' x O P(√3, 1)
  2. On comparing z with the standard polar form of a complex number, we get,
    r cos θ=3 and r sin θ=1
    Now, r cos θ=3(1)r2 cos2θ=32(2)r sinθ=1(3)r2 sin2θ=12(4) On Adding (2) and (4) we get,
    r2 cos2θ+r2 sin2θ=32+12r2(cos2θ+sin2θ)=3+1r2=4[Since, cos2θ+sin2θ=1]r=2[Conventionally r>0]
  3. Substituting the value of r in eq (1) and (3) we get,
    cosθ=32 and sinθ=12
    θ=π6
  4. Hence, the polar form of the complex number z=3+i is 2(cosπ6+i sinπ6).

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