Problem 51 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ \sqrt{2}+\sqrt{2} i $$

Step-by-Step Solution

Answer

The polar form is \( 2(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) \).

Step 1: Identify real and imaginary parts

The given complex number is \( \sqrt{2} + \sqrt{2}i \) where the real part, \( a \), is \( \sqrt{2} \) and the imaginary part, \( b \), is \( \sqrt{2} \).

Step 2: Calculate the modulus of the complex number

The modulus \( r \) of a complex number \( a + bi \) is given by \( r = \sqrt{a^2 + b^2} \). Substitute \( a = \sqrt{2} \) and \( b = \sqrt{2} \): \[ r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \].

Step 3: Determine the argument \( \theta \)

The argument \( \theta \) is given by \( \tan^{-1}(b/a) \). Compute: \[ \tan^{-1}(\sqrt{2}/\sqrt{2}) = \tan^{-1}(1) = \frac{\pi}{4} \].

Step 4: Write the complex number in polar form

The polar form of a complex number is \( r(\cos \theta + i\sin \theta) \). From previous steps, \( r = 2 \) and \( \theta = \frac{\pi}{4} \). Therefore, the polar form is \[ 2(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) \].

Key Concepts

Modulus of a Complex Number

When dealing with complex numbers, finding the modulus is a fundamental step in understanding their distance from the origin in the complex plane. The modulus of a complex number, often denoted as \(|z|\) for a complex number \(z = a + bi\), is calculated using the formula: \[ r = \sqrt{a^2 + b^2} \] where \(a\) is the real part and \(b\) is the imaginary part of the complex number. In our exercise, we have a complex number \( \sqrt{2} + \sqrt{2}i \). Here, both the real part, \(a\), and the imaginary part, \(b\), are equal to \(\sqrt{2}\). To find the modulus \(r\): - Square both parts: \((\sqrt{2})^2 = 2\) - Add them: \(2 + 2 = 4\) - Take the square root: \(\sqrt{4} = 2\) The modulus of this complex number is \(r = 2\). This means the complex number is 2 units away from the origin on the complex plane.

Argument of a Complex Number

The argument of a complex number is essential for converting a complex number into polar form. It represents the angle \(\theta\) that the complex number makes with the positive real axis. To find the argument \(\theta\) of a complex number \(z = a + bi\), you use the formula: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] In our example, both the real part \(a = \sqrt{2}\) and imaginary part \(b = \sqrt{2}\) are the same. Thus, we calculate: - Ratio: \(\frac{b}{a} = \frac{\sqrt{2}}{\sqrt{2}} = 1\) - Inverse tangent: \(\tan^{-1}(1) = \frac{\pi}{4}\) This results in the argument \(\theta = \frac{\pi}{4}\), which tells us that the angle is \(45^{\circ}\). This angle indicates the direction of the complex number from the origin, pointing diagonally in the plane.

Trigonometry in Polar Coordinates

Trigonometry plays a crucial role when expressing complex numbers in polar form. The polar form can be very useful, especially when performing multiplication and division, or raising numbers to powers. The polar form of a complex number \(z = a + bi\) is given by: \[ r(\cos \theta + i\sin \theta) \] Where: - \(r\) is the modulus of the complex number. - \(\theta\) is the argument, which we've already determined. In this exercise, we found: - Modulus \(r = 2\) - Argument \(\theta = \frac{\pi}{4}\) Therefore, substituting these values gives: \[ 2\left(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right) \] This form utilizes trigonometric functions to represent the location and direction of the complex number in the plane. It's compact and emphasizes both the distance from the origin (radius \(r\)) and the angle \(\theta\) around it, seamlessly blending geometry with algebra.