_ Write −√3 + i in polar form.

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Identify the complex number given: \(-\sqrt{3} + i\). Here, the real part is \(-\sqrt{3}\) and the imaginary part is \(1\).

Calculate the modulus \(r\) of the complex number using the formula \(r = \sqrt{x^2 + y^2}\), where \(x\) is the real part and \(y\) is the imaginary part. So, compute \(r = \sqrt{(-\sqrt{3})^2 + 1^2}\).

Find the argument \(\theta\) (also called the angle) using \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Substitute \(x = -\sqrt{3}\) and \(y = 1\) to get \(\theta = \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right)\).

Determine the correct quadrant for the angle \(\theta\). Since the real part is negative and the imaginary part is positive, the complex number lies in the second quadrant. Adjust the angle accordingly to reflect this quadrant.

Write the polar form of the complex number as \(r(\cos \theta + i \sin \theta)\) or equivalently \(r e^{i \theta}\), using the modulus \(r\) and the argument \(\theta\) found in the previous steps.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers in Cartesian and Polar Form

A complex number can be expressed in Cartesian form as a + bi, where a is the real part and b is the imaginary part. The polar form represents the same number using a magnitude (radius) and an angle (argument), written as r(cos θ + i sin θ) or r e^{iθ}, which is useful for multiplication and division.

Magnitude (Modulus) of a Complex Number

The magnitude of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, calculated as r = √(a² + b²). This value represents the radius in the polar form and is always non-negative.

Argument (Angle) of a Complex Number

The argument θ of a complex number is the angle formed with the positive real axis, found using θ = arctan(b/a). It determines the direction of the vector representing the complex number in the plane and is essential for expressing the number in polar form.

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