Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -2-2 i $$

Step-by-Step Solution

Answer

The complex number in trigonometric form is \(2\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(-2 - 2i.\) Here, the real part is \(a = -2\) and the imaginary part is \(b = -2\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by the formula: \[r = \sqrt{a^2 + b^2} \] Substitute the values \(a = -2\) and \(b = -2\):\[r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\].

Step 3: Determine the Argument

The argument \(\theta\) is found using the formula:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]Substitute \(b = -2\) and \(a = -2\):\[\theta = \tan^{-1}\left(\frac{-2}{-2}\right) = \tan^{-1}(1) = \frac{\pi}{4}\].However, this angle is in the first quadrant. Since both real and imaginary parts are negative, it should be in the third quadrant. Add \(\pi\) to put it in the correct quadrant:\[\theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}\].

Step 4: Write in Trigonometric Form

The trigonometric form of a complex number is\[r(\cos \theta + i \sin \theta)\].Now substitute \(r = 2\sqrt{2}\) and \(\theta = \frac{5\pi}{4}\):\[2\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})\].

Key Concepts

Trigonometric Form

To express a complex number in trigonometric form, we transform its standard form, which is generally given as \(a + bi\), into a polar representation. This form highlights the magnitude and the argument of the complex number, making certain operations, like multiplication and division, more convenient.

### Transforming to Trigonometric FormThe trigonometric form of a complex number is written as:

• \[r(\cos \theta + i \sin \theta)\]

In this form:

• \(r\) is the magnitude (or modulus) of the complex number.
• \(\theta\) is the argument of the complex number, indicating the angle of the number in relation to the positive x-axis.

By converting the complex number \(-2 - 2i\) into this form, we identify both its magnitude and direction accurately. This is useful particularly in graphical or plotting contexts, as well as simplifying computations like finding roots or powers of complex numbers.

Magnitude of Complex Numbers

The magnitude of a complex number tells us how 'far' the number is from the origin in the complex plane. It's a valuable part of its polar form.

### Calculating the MagnitudeTo find the magnitude, use the formula:

• \[r = \sqrt{a^2 + b^2}\]

When applied to a complex number \(a + bi\), you're essentially using the Pythagorean theorem to calculate the length of the diagonal in a right-angle triangle formed by its components.

In this case, our complex number is \(-2 - 2i\), so:

• The real part \(a = -2\).
• The imaginary part \(b = -2\).
• The magnitude becomes \[\sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}\]

Finding the magnitude is an essential first step to expressing any complex number in trigonometric form, as it directly influences the values in the final expression.

Argument of a Complex Number

The argument of a complex number details the directional angle from the positive x-axis in the complex plane. This angle is crucial for depicting the number's specific position.

### Calculating the ArgumentTo find the argument \(\theta\), apply the formula:

• \[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]

For \(-2 - 2i\), substituting in the values, we initially get:

• \[\theta = \tan^{-1}\left(\frac{-2}{-2}\right) = \tan^{-1}(1) = \frac{\pi}{4}\]

However, this represents an angle in the first quadrant. Both components being negative suggests the number resides in the third quadrant, requiring us to shift it by adding \(\pi\):

• \[\theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}\]

Correctly determining this angle allows us to place the complex number accurately within the trigonometric unit circle.