_Write −√3 + i in polar form.

Verified step by step guidance

1. Identify the real part \( a = -\sqrt{3} \) and the imaginary part \( b = 1 \) of the complex number.
2. Calculate the modulus \( r \) using the formula \( r = \sqrt{a^2 + b^2} \).
3. Determine the argument \( \theta \) using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Since the real part is negative and the imaginary part is positive, the angle is in the second quadrant.
4. Express the complex number in polar form as \( r(\cos \theta + i\sin \theta) \) or \( re^{i\theta} \).
5. Substitute the values of \( r \) and \( \theta \) into the polar form expression.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in the plane using a distance from the origin and an angle from the positive x-axis. In polar form, a complex number is expressed as r(cos θ + i sin θ), where r is the modulus (distance) and θ is the argument (angle). This representation is particularly useful for multiplication and division of complex numbers.

Modulus of a Complex Number

The modulus of a complex number a + bi, denoted as |z|, is calculated using the formula |z| = √(a² + b²). It represents the distance of the point (a, b) from the origin in the complex plane. For the complex number −√3 + i, the modulus helps determine the radial distance needed for its polar representation.

Argument of a Complex Number

The argument of a complex number is the angle θ formed with the positive x-axis, calculated using the arctangent function: θ = arctan(b/a). It indicates the direction of the complex number in the polar coordinate system. For −√3 + i, the argument must be determined considering the quadrant in which the point lies, which is essential for accurate polar representation.