Represent z=1+i √3 in the polar form.

Question

Represent z=1+i √3 in the polar form. (A)  cos (π/3)+i sin (π/3) (B)  cos (π/3)-i sin (π/3) (C) 2( cos (π/3)+ i sin (π/3)) (D) 4( cos (π/3)+i sin (π/3))

Tardigrade Admin · Accepted Answer

Let 1=r cos θ, √3=r sin θ By squaring and adding, we get r2( cos 2 θ+ sin 2 θ)=4 i.e, r=√4=2 Therefore, cos θ=(1/2), sin θ=(√3/2), which gives θ=(π/3) Therefore, required polar form is: z=2( cos (π/3)+i sin (π/3))

Q. Represent $z = 1 + i 3$ in the polar form.

$cos \frac{π}{3} + i sin \frac{π}{3}$

$cos \frac{π}{3} - i sin \frac{π}{3}$

$2 (cos \frac{π}{3} + i sin \frac{π}{3})$

$4 (cos \frac{π}{3} + i sin \frac{π}{3})$

Let $1 = r cos θ, 3 = r sin θ$
By squaring and adding, we get
$r^{2} (cos^{2} θ + sin^{2} θ) = 4$
i.e, $r = 4 = 2$
Therefore, $cos θ = \frac{1}{2}, sin θ = \frac{3}{2}$ ,
which gives $θ = \frac{π}{3}$
Therefore, required polar form is :
$z = 2 (cos \frac{π}{3} + i sin \frac{π}{3})$

Q. Represent z=1+i3​ in the polar form.

cos3π​+isin3π​

cos3π​−isin3π​

2(cos3π​+isin3π​)

4(cos3π​+isin3π​)