Problem 47 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\left[\frac{5}{3}\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\right]\left[\frac{2}{3}\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\right]$$

Step-by-Step Solution

Answer

The result is \(\frac{10}{9}( \cos \frac{5π}{6} + i \sin \frac{5π}{6})\)

Step 1: Convert angles to radians

Convert all angles from degrees to radians. Find the radian equivalent of 120 degrees and 30 degrees using the formula \(Radian = Degree \times \frac{π}{180}\) yielding \( \frac{2π}{3} \) and \(\frac{π}{6} \) respectively.

Step 2: Multiplication of Magnitudes

Multiply the magnitudes (the real numbers in front of the brackets). In this case, multiply \( \frac{5}{3} \) by \( \frac{2}{3} \) to get \( \frac{10}{9} \) as the new magnitude.

Step 3: Addition of Angles

Add the angles. Here, add \(\frac{2π}{3} \) and \(\frac{π}{6} \) to get \(\frac{5π}{6} \) as the new angle.

Step 4: Representation in Trigonometric Form

Finally, represent the result in the trigonometric form. The result will then be \(\frac{10}{9}( \cos \frac{5π}{6} + i \sin \frac{5π}{6})\)

Key Concepts

Trigonometric Form

Complex numbers can be expressed in what's known as trigonometric form. This is sometimes called polar form as well. Instead of expressing a complex number as a sum of its real and imaginary parts, we express it in terms of a magnitude (or modulus) and an angle (called the argument). The general formula is \[ r(\cos \theta + i \sin \theta) \] where "r" is the magnitude of the complex number, and \( \theta \) is the angle in radians. In this way of expressing numbers, multiplication and division are made much simpler, as we'll see in the other steps.

Radians Conversion

To work with complex numbers in trigonometric form, it's crucial to handle angles in radians rather than degrees. This is because most mathematical functions—especially those involving calculus and trigonometry—use radian measure.

• Convert an angle from degrees to radians using the formula: \\[ \text{Radian} = \text{Degree} \times \frac{\pi}{180} \]
• For example, converting 120 degrees to radians: \\[ 120 \times \frac{\pi}{180} = \frac{2 \pi}{3} \]
• Similarly, 30 degrees into radians: \\[ 30 \times \frac{\pi}{180} = \frac{\pi}{6} \]

By expressing angles in radians, all calculations become consistent throughout mathematical applications.

Multiplication of Magnitudes

When multiplying complex numbers in trigonometric form, the magnitudes (or moduli) of the numbers are multiplied directly. This is much simpler than multiplying two binomials if the complex numbers were in rectangular form.

• Take the magnitudes or the coefficients outside the trigonometric expressions.
• Multiply them directly: \\[ \frac{5}{3} \times \frac{2}{3} = \frac{10}{9} \]

This approach benefits from being straightforward, reducing the chances of errors that could arise if the numbers were multiplied in standard form. It provides a single number for the modulus of the product of two complex numbers.

Addition of Angles

After determining the product of the magnitudes, the next step in multiplying two complex numbers in trigonometric form is the addition of their angles. By using the angles in their radian form, the addition becomes straightforward.

• Simply add the two angles from each complex number to find the resultant angle: \\[ \frac{2\pi}{3} + \frac{\pi}{6} = \frac{5\pi}{6} \]

This summed angle, along with the multiplied magnitudes, forms the new trigonometric expression for the product of the two complex numbers. It's especially convenient when visualizing or plotting complex numbers on the complex plane.