Problem 108 - Precalculus with Limits

Question

Quotient of Two Complex Numbers Given two complex numbers \(z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)\) and \(z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right), z_{2} \neq 0,\) show that $$\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]$$

Step-by-Step Solution

Answer

The expression \( \frac{z_{1}}{z_{2}} \) can indeed be simplified to \( \frac{r_{1}}{r_{2}} [ \cos ( \theta_{1} - \theta_{2} )+ i \sin ( \theta_{1} - \theta_{2} ) ] \), given that \( z_{1}=r_{1}(\cos \theta_{1}+i \sin \theta_{1}) \) and \( z_{2}=r_{2}(\cos \theta_{2}+i \sin \theta_{2}) \), and \( z_{2} \neq 0 \). This proofs the division property of complex numbers in their polar form.

Step 1: Simplify left-side expression

Start with the left-hand side of the equation which is \( \frac{z_{1}}{z_{2}} \). This is the expression we will work on and try to simplify it to look like the right-side expression. This can be rewritten as \( \frac{r_{1}(\cos \theta_{1}+i \sin \theta_{1})}{r_{2}(\cos \theta_{2}+i \sin \theta_{2})} \).

Step 2: Separate real and imaginary parts

Write the equation from Step 1 in the format of \( A + Bi \) where A is the real part and Bi is the imaginary part of the expression. This can be done by multiplying both numerator and denominator by the conjugate of the denominator. The resulting equation will be \( \frac{r_{1}r_{2}[(\cos \theta_{1} \cos \theta_{2} + \sin \theta_{1} \sin \theta_{2}) + i( \sin \theta_{1} \cos \theta_{2} - \cos \theta_{1} \sin \theta_{2})] }{r_{2}^{2}} \)

Step 3: Apply trigonometric identities

Now transform the expression by applying some trigonometric rules. Use the identity of cosine of difference and sine of difference, where \( \cos (A - B ) = \cos A \cos B + \sin A \sin B\) and \( \sin (A - B ) = \sin A \cos B - \cos A \sin B\). So it becomes \( \frac{r_{1}}{r_{2}} [ \cos ( \theta_{1} - \theta_{2} )+ i \sin ( \theta_{1} - \theta_{2} ) ] \)

Step 4: Final verification

The expression found in step 3 is the same as the right-hand side of the given equation.

Key Concepts

Complex Number Division

Dividing complex numbers can often seem daunting at first, but breaking the process down step by step can make it easier to understand, especially when the numbers are in polar form. When we divide two complex numbers, we want to find the quotient of one by the other, represented in a format that is easy to analyze.

Given two complex numbers in polar form, division will involve the moduli and arguments of these numbers separately. We start with the formula:

• Numerator: \( z_1 = r_1(\cos \theta_1 + i \sin \theta_1) \)
• Denominator: \( z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \)

To divide, simplify using moduli: \( \frac{r_1}{r_2} \), and subtract arguments: \( \theta_1 - \theta_2 \). This process makes use of the identity properties of complex numbers and polar coordinates, ensuring the operation remains within the realm of polar form.

Trigonometric Identities

To simplify expressions during division of complex numbers, especially in polar form, trigonometric identities become crucial tools. In this exercise, the cosine and sine difference identities play a central role. These identities help in transforming the products of trigonometric functions into a more manageable form.

Let's consider the identities:

• The cosine of a difference: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
• The sine of a difference: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

These identities provide a direct pathway to rewrite complicated expressions in a significantly simpler form. They achieve this by converting products into sums and differences, a process essential for obtaining the solution as shown in the step-by-step solution. Applying these identities correctly ensures that division in the realm of polar form maintains mathematical integrity and simplicity.

Polar Form of Complex Numbers

The polar form of complex numbers is highly beneficial for operations like multiplication and division. When a complex number is expressed in polar form, it is represented by its magnitude (or modulus) and its angle (or argument). This representation simplifies many calculations by converting complex operations into simpler algebraic manipulations.

In the expression \( z = r(\cos \theta + i \sin \theta) \),

• \( r \) denotes the magnitude, which is the distance from the origin to the point, equivalent to the modulus of the number
• \( \theta \) is the angle, representing the argument of the complex number

For division, polar form allows dividing the magnitudes and subtracting the angles. The neat separation between magnitude and direction makes the polar system an elegant choice for many complex number operations. This transformation into polar form not only simplifies division but also fits naturally into the framework of trigonometric identities, making solutions both precise and straightforward.