Problem 51 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\frac{3\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)}{9\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)}$$

Step-by-Step Solution

Answer

The division of the given complex numbers in trigonometric form is: \(0.333 * (\cos(30) + i \sin(30))\).

Step 1: Identify the modulus and the argument

The division of two complex numbers in trigonometric form is given by: \(\frac{r_1 (\cos(\theta_1)+i \sin(\theta_1))}{r_2 (\cos(\theta_2)+i \sin(\theta_2))}\), where \(r_1\) and \(r_2\) are the modulus and \(\theta_1\) and \(\theta_2\) are the argument for the numerator and denominator complex numbers respectively. In this case, \(r_1 = 3, \theta_1 = 50\), for the numerator and \(r_2 = 9, \theta_2 = 20\), for the denominator.

Step 2: Divide the modulus and subtract the arguments

To divide the complex numbers, we divide the modulus and subtract the argument of the denominator from the numerator. This gives us: \(\frac{r_1}{r_2}\) and \(\theta_1 - \theta_2\). Hence, \(\frac{r_1}{r_2} = \frac{3}{9} = 0.333\), and \(\theta_1 - \theta_2 = 50 - 20 = 30\) degrees.

Step 3: Insert the values into the trigonometric form

Now, we can write down our result in trigonometric form as: \(\frac{r_1}{r_2} * (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))\). Substituting the calculated values, the division of the complex numbers is: \(0.333 * (\cos(30) + i \sin(30))\)

Key Concepts

Trigonometric Form

Complex numbers can be expressed in several forms, with the trigonometric form being particularly useful for multiplication and division. The trigonometric form of a complex number uses the modulus (or absolute value) and argument (or angle) of the number. This form is represented as: \[ r(\cos(\theta) + i\sin(\theta)) \] Where:

• \( r \) is the modulus, which denotes the distance from the origin to the point in the complex plane.
• \( \theta \) is the argument, which signifies the angle made with the positive real axis.

This representation becomes incredibly handy because it transforms complex number operations into simple algebraic operations on real numbers, specifically multiplication and addition, for the modulus and argument, respectively.

Modulus and Argument

Understanding the modulus and argument is crucial when working with complex numbers in trigonometric form. The modulus \( r \) gives the magnitude or length of the complex vector, calculated as the square root of the sum of the squares of its real and imaginary parts. Meanwhile, the argument \( \theta \) specifies the direction and is given by the arctangent of the imaginary part over the real part. In the context of division: - Divide the modulus of the numerator by the modulus of the denominator.- Subtract the argument of the denominator from the argument of the numerator. With this systematic approach, we transform complicated operations on complex numbers into straightforward tasks.

Division of Complex Numbers

Dividing complex numbers in trigonometric form requires careful handling of both the modulus and the argument. The operation can be summarized in a few steps: 1. **Modulus Division:** Divide the modulus of the numerator by the modulus of the denominator. This operation reflects the magnitude aspect of the division.2. **Argument Subtraction:** Subtract the argument of the denominator from the argument of the numerator, simplifying the angle component of the operation.For example, given: \[ \frac{3(\cos 50^\circ + i \sin 50^\circ)}{9(\cos 20^\circ + i \sin 20^\circ)} \] We calculate:- Modulus: \( \frac{3}{9} = 0.333 \)- Argument: \( 50^\circ - 20^\circ = 30^\circ \)The result is: \[ 0.333(\cos 30^\circ + i \sin 30^\circ) \] This concise expression showcases the power and simplicity of using trigonometric form for complex number division.