Write the complex number into polar form z = 1 + sqrt 3i

Question

asked Feb 12, 2024 91.1k views

2 Answers

Nhat Dinh · Answer 1 · 2024-02-13T19:35:48+0000

Final answer:

The complex number
1 + √(3)i in polar form is
$2(cos((\pi)/(3)) + i sin((\pi)/(3)))$ .

Step-by-step explanation:

To write the given complex number in polar form, we first need to express the complex number as

z = a + bi,

where

a is the real part and

bi is the imaginary part.

In this case, we have
z = 1 + √(3)i.

The polar form of a complex number is represented as r(cos(θ) + i sin(θ)),

where

r is the magnitude (modulus) of the complex number and

θ is the argument (angle).

To find r, we calculate the magnitude of the complex number using the formula

r = √(a^2 + b^2)

In this case,
$r = \sqrt{1^2 + (√(3))^2} = 2$ .

To find θ, we use the arctangent function (tan^-1(b/a)) to find the angle.

Since a = 1 and
b = √(3)

θ
= tan^-1(√(3)/1) ,

which gives us

θ =
$(\pi)/(3)$

So, the polar form of the complex number
z = 1 + √(3)i is
$z = 2(cos((\pi)/(3)) + i sin((\pi)/(3)))$ .