Problem 50 - Precalculus with Limits

Question

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \([\frac{1}{2}(\cos\ 100^{\circ} + i\ \sin\ 100^{\circ})][\frac{4}{5}(\cos\ 300^{\circ} + i\ \sin\ 300^{\circ})]\)

Step-by-Step Solution

Answer

The result is \(\frac{2}{5}(\cos 40^{\circ} + i\ \sin 40^{\circ})\)

Step 1 - Multiply the magnitudes

Multiply the magnitudes (\(r\) values) of the two complex numbers: \(\frac{1}{2} \times \frac{4}{5} = \frac{2}{5}\)

Step 2 - Add the angles

Add the angles (\(θ\) values) of the two complex numbers: \(100^{\circ} + 300^{\circ} = 400^{\circ}\). Since a full circle is \(360^{\circ}\), subtracting \(360^{\circ}\) leaves us with \(40^{\circ}\)

Step 3 - Write the result in trigonometric form

Combine the results from steps 1 and 2 to write the result in trigonometric form, i.e., \(\frac{2}{5}(\cos 40^{\circ} + i\sin 40^{\circ})\)

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts – a real part and an imaginary part. The imaginary part is expressed using the imaginary unit, which is represented by the letter \(i\), where \(i\) is equal to the square root of \(-1\). Complex numbers are often depicted in the form \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component. Using trigonometric form, however, complex numbers can be expressed as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude of the complex number and \(\theta\) is the angle it makes with the positive x-axis. This is particularly useful when performing multiplication or division, as it simplifies the process dramatically. In trigonometric form, the focus is on the magnitude and the angle rather than individual real and imaginary parts.

Magnitude Multiplication

In trigonometric form, multiplication of complex numbers involves multiplying their magnitudes. If you have two complex numbers represented as \(r_1 (\cos \theta_1 + i\sin \theta_1)\) and \(r_2 (\cos \theta_2 + i\sin \theta_2)\), their product will have a magnitude of \(r_1 \cdot r_2\). This is because the magnitudes essentially scale the size of the vector represented by the complex number. For example, multiplying the magnitudes \(\frac{1}{2}\) and \(\frac{4}{5}\) yields \(\frac{2}{5}\). The resultant magnitude gives the size of the new complex number after multiplication.

Angle Addition

Angle addition in the context of complex numbers in trigonometric form is about combining the angles of the two numbers being multiplied. Each complex number forms an angle with the positive x-axis, known as its argument. When multiplying two complex numbers, their angles are simply added together. For instance, adding angles \(100^{\circ}\) and \(300^{\circ}\) gives \(400^{\circ}\). Since angles are cyclical and \(360^{\circ}\) forms a circle, you subtract \(360^{\circ}\) from \(400^{\circ}\) to find an equivalent angle that places the number within the standard range of \(0\) to \(360\). Here, this adjustment gives the final angle of \(40^{\circ}\). This new angle, combined with the magnitude, completes the trigonometric representation of the resultant complex number.