Question

Question 7 Convert the given complex number in polar form: 3^(1/2) + i

Class X1 - Maths -Complex Numbers and Quadratic Equations Page 108

Answer 1

Let z = √3 + i
we know, any complex number in the polar form is r(cos∅ + i sin∅) .
so,
√3 + i = r(cos∅ + isin∅)
√3 + i = rcos∅ +i rsin∅
rcos∅ = √3 _____________(1)
rsin∅ = 1 _____________(2)
square and add equations (1) and (2)
r²( sin²∅ + cos²∅) = √3² + 1²
r².1 = 3 + 1 [ we know, sin²∅ + cos²∅ =1]
r = ±2
but distance doesn't negative.
so, r = 2

divide (2) ÷ (1)
sin∅/cos∅ = 1/√3
tan∅ = tan(π/6)
∅ = π/6
because ∅ lies on 1st quadrant so,
arg(z) = ∅
= π/6
thus,
z = √3 + i= 2[cos(π/6)+isin(π/6) ]