Problem 49 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)\left(\cos 330^{\circ}+i \sin 330^{\circ}\right)$$

Step-by-Step Solution

Answer

The result in trigonometric form of the multiplication operation is \(\cos(50^\circ) + i\sin(50^\circ)\).

Step 1: Use the multiplication rule for complex numbers

The multiplication rule for complex numbers states that the product of two complex numbers in trigonometric form is \(\cos(\alpha+\beta) + i\sin(\alpha+\beta)\), where \(\alpha\) and \(\beta\) are the angles of the two complex numbers. Here, \(\alpha = 80^\circ\) and \(\beta = 330^\circ\). So, the product would be \(\cos(80^\circ + 330^\circ) + i\sin(80^\circ + 330^\circ)\).

Step 2: Compute the sum of the angles

The sum of angles \(\alpha\) and \(\beta\) would be \(80^\circ + 330^\circ = 410^\circ\).

Step 3: Normalize the angle

Since the standard range within which an angle should lie is between \(0^\circ\) and \(360^\circ\), normalize \(410^\circ\) by subtracting \(360^\circ\) until the result is within this range. \(410^\circ - 360^\circ = 50^\circ\).

Step 4: Substitute the normalized angle

Substitute the normalized angle back into the equation, yielding \(\cos(50^\circ) + i\sin(50^\circ)\).

Key Concepts

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number is a unique way to represent complex numbers using trigonometric functions. Instead of writing a complex number as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, we use angles and magnitudes in this representation. This form is particularly useful in operations like multiplication and division.

In trigonometric form, a complex number is expressed as \(r(\cos(\theta) + i\sin(\theta))\). Here:

• \(r\) is the magnitude of the complex number, equivalent to the distance from the origin on the complex plane, calculated as \(r = \sqrt{a^2 + b^2}\).
• \(\theta\) is the argument or angle, measured in degrees or radians, representing the direction of the complex number from the positive real axis.

The trigonometric form is intuitive in expressing rotations and scaling in the complex plane, which simplifies complexities involved in multiplication and division.

Multiplication of Complex Numbers in Trigonometric Form

Multiplying complex numbers in their trigonometric form simplifies the operation significantly, avoiding lengthy algebraic expansions. The product of two complex numbers, each in trigonometric form, is found by simply multiplying their magnitudes and adding their angles.

Given two complex numbers \(z_1 = r_1(\cos \alpha + i\sin \alpha)\) and \(z_2 = r_2(\cos \beta + i\sin \beta)\), their product \(z_1 \times z_2\) is represented as:

• \(z_1 \times z_2 = r_1r_2(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\).

This operation reflects not only the multiplication of sizes but also compounding their directional angles. In our exercise, by multiplying \(\cos 80^\circ + i\sin 80^\circ\) and \(\cos 330^\circ + i\sin 330^\circ\), the angles \(80^\circ\) and \(330^\circ\) sum up to \(410^\circ\). Consequently, you can express the product directly based on this new angle.

Normalizing Angles in Trigonometric Form

After adding the angles when multiplying complex numbers, often, the resulting angle may exceed the standard range of \(0^\circ\) to \(360^\circ\). Normalizing these angles ensures the complex number remains consistent and correctly represented in standard form.

To normalize an angle, you adjust it to fit within the desired range by removing full rotations, which are \(360^\circ\) or \(2\pi\) radians. This is simply done by subtracting \(360^\circ\) from any angle greater than \(360^\circ\) until it falls below \(360^\circ\).

• For example, an angle of \(410^\circ\) is normalized by subtracting \(360^\circ\), yielding \(50^\circ\).

This normalized angle can then be used in the trigonometric representation of the complex number, ensuring accuracy and adherence to standard mathematical conventions throughout calculations.