Problem 58 - Contemporary Precalculus

Question

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$12\left(\cos \frac{11 \pi}{12}+i \sin \frac{11 \pi}{12}\right) \cdot \frac{7}{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

Question: Multiply the following complex numbers in polar form and express the result in both polar and rectangular form: $$ z_1 = 12\left(\cos \frac{11 \pi}{12}+i \sin \frac{11 \pi}{12}\right) \quad \text{and} \quad z_2 = \frac{7}{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) $$ Answer: After performing the multiplication, we have: Polar form: $$ z = 42\left(\cos\left(\frac{14\pi}{12}\right)+i \sin\left(\frac{14\pi}{12}\right)\right) $$ Rectangular form: $$ z = -21\sqrt{3} -21i $$

Step 1: Identify the polar form of given complex numbers

The given complex numbers are: $$ z_1 = 12\left(\cos \frac{11 \pi}{12}+i \sin \frac{11 \pi}{12}\right) \quad \text{and} \quad z_2 = \frac{7}{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) $$

Step 2: Use polar form formula for multiplication

To multiply two complex numbers in polar form, we multiply their magnitudes and add their angles. The formula for multiplication is: $$ z_1 \cdot z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) $$

Step 3: Perform the multiplication

Using the given complex numbers and the multiplication formula, we have: $$ z_1 \cdot z_2 = 12\left(\frac{7}{2}\right)\left(\cos\left(\frac{11 \pi}{12} +\frac{\pi}{4}\right)+i \sin\left(\frac{11 \pi}{12} +\frac{\pi}{4}\right)\right) $$

Step 4: Simplify the polar form

Now let's simplify the magnitude and the angle in the polar form: $$ z_1 \cdot z_2 = 42\left(\cos\left(\frac{14 \pi}{12}\right)+i \sin\left(\frac{14\pi}{12}\right)\right) $$

Step 5: Convert to rectangular form

To convert the polar form to rectangular form, we use the following formulas: $$ a = r\cos\theta \quad \text{and} \quad b = r\sin\theta $$ Applying these formulas to our simplified polar form: $$ a = 42\cos\left(\frac{14 \pi}{12}\right) \quad \text{and} \quad b = 42\sin\left(\frac{14 \pi}{12}\right) $$

Step 6: Calculate the rectangular form

Finally, let's find the value of 'a' and 'b' and write the complex number in rectangular form: $$ a = 42 \cos\left(\frac{7\pi}{6}\right) = -42\left(\frac{\sqrt{3}}{2}\right) = -21\sqrt{3} $$ $$ b = 42 \sin\left(\frac{7\pi}{6}\right) = -42\left(\frac{1}{2}\right) = -21 $$ So the rectangular form of the complex number is: $$ z = -21\sqrt{3} -21i $$

Step 7: Present the final answer

The final answer after multiplication in both polar and rectangular form is: Polar form: $$ z = 42\left(\cos\left(\frac{14\pi}{12}\right)+i \sin\left(\frac{14\pi}{12}\right)\right) $$ Rectangular form: $$ z = -21\sqrt{3} -21i $$

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be represented in many ways, with the polar form being exceptionally useful in certain mathematical operations, including multiplication and division. A complex number in polar form is expressed as \( r(\text{cos} \theta + i \text{sin} \theta) \), where \( r \) is the magnitude (also known as the modulus) and \( \theta \) is the angle (also known as the argument) formed with the positive real axis.

The polar form allows us to interpret a complex number as a point in the complex plane that can be defined by a distance from the origin (\( r \)) and a direction (\( \theta \)). For example, the polar form of the number \( 12(\text{cos} \frac{11 \pi}{12} + i \text{sin} \frac{11 \pi}{12}) \) signifies a vector 12 units long, pointing in a direction that makes an angle of \( \frac{11 \pi}{12} \) radians with the positive real axis.

This representation is particularly powerful in simplifying the multiplication and division of complex numbers, as it converts these operations into simple addition or subtraction of angles and multiplication or division of magnitudes, which will be further explained in subsequent sections.

Rectangular Form of Complex Numbers

The rectangular form of a complex number is likely the one most students are familiar with. It is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number. This form is akin to Cartesian coordinates and is useful for visualization on the complex plane, where the real part corresponds to the x-coordinate and the imaginary part to the y-coordinate.

For instance, if we have a complex number \( z = -21\sqrt{3} - 21i \), its real part is \( -21\sqrt{3} \) and its imaginary part is \( -21 \). These values tell us that the number lies 21\sqrt{3} units to the left and 21 units down from the origin in the complex plane.

The rectangular form is very convenient for addition and subtraction of complex numbers, akin to adding or subtracting vectors. When looking at the intersection of algebra and geometry, this form can also facilitate understanding of concepts like the conjugate of a complex number and the modulus or absolute value.

Multiplying Complex Numbers in Polar Form

Multiplying complex numbers becomes straightforward when they are expressed in polar form. The multiplication of two complex numbers in polar form requires us to multiply their magnitudes \( r_1 \) and \( r_2 \), and add their angles \( \theta_1 \) and \( \theta_2 \). The product is a new complex number with a magnitude equal to the product of the original magnitudes and an angle equal to the sum of the original angles.

For example, to multiply \( 12(\text{cos} \frac{11 \pi}{12} + i \text{sin} \frac{11 \pi}{12}) \) and \( \frac{7}{2}(\text{cos} \frac{\pi}{4} + i \text{sin} \frac{\pi}{4}) \), we simply calculate the new magnitude by multiplying 12 and \( \frac{7}{2} \) to get 42, and add the angles \( \frac{11 \pi}{12} \) and \( \frac{\pi}{4} \) to find the new angle \( \frac{14 \pi}{12} \). The result is the polar form of the product:\

• Magnitude: \( r = 12 \times \frac{7}{2} = 42 \)
• Angle: \( \theta = \frac{11 \pi}{12} + \frac{\pi}{4} = \frac{14 \pi}{12} \)

Thus, the product in polar form is \( 42(\text{cos} \frac{14 \pi}{12} + i \text{sin} \frac{14 \pi}{12}) \). This rule significantly simplifies the multiplication of complex numbers compared to the rectangular form, where the process involves distributive property and combining like terms.

Converting Polar to Rectangular Form

Converting a complex number from polar to rectangular form involves the use of trigonometric identities. The conversion utilizes the formulae \( a = r \cdot \text{cos}(\theta) \) and \( b = r \cdot \text{sin}(\theta) \), where \( a \) and \( b \) will represent the real and imaginary components respectively in the rectangular form \( a + bi \).

Let's take our product of complex numbers in polar form, \( 42(\text{cos} \frac{14 \pi}{12} + i \text{sin} \frac{14 \pi}{12}) \), and convert it to rectangular form. We find the values of \( a \) and \( b \) by applying the trigonometric functions:\

• Real part: \( a = 42 \cdot \text{cos}(\frac{14 \pi}{12}) \) which simplifies to \( -21\sqrt{3} \)
• Imaginary part: \( b = 42 \cdot \text{sin}(\frac{14 \pi}{12}) \) which reduces to \( -21 \)

Therefore, the rectangular form of the complex number is \( -21\sqrt{3} - 21i \). This conversion is essential when we need to perform operations that are more naturally done in the rectangular form, such as adding complex numbers or finding their conjugal pairs, but starting with coordinates given in polar form.