Problem 57 - 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 \(a+b i\). $$3\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \cdot 12\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right)$$

Step-by-Step Solution

Answer

Answer: The product of the two complex numbers in polar form is: $$36\left(cos \frac{\pi}{2} + i\sin \frac{\pi}{2}\right)$$ In rectangular form, the product is: $$0 + 36i$$

Step 1: Multiply complex numbers in polar form

Using the formula for multiplication of complex numbers in polar form: $$(r_1(\cos \theta_1 + i\sin \theta_1))\cdot (r_2(\cos \theta_2 + i\sin\theta_2)) = (r_1\cdot r_2)((\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2))$$ We have: - \(r_1 = 3\), \(\theta_1 = \frac{\pi}{8}\) - \(r_2 = 12\), \(\theta_2 = \frac{3\pi}{8}\) Let's apply the formula with these values: $$(3(\cos \frac{\pi}{8} + i\sin \frac{\pi}{8}))(12(\cos \frac{3\pi}{8} + i\sin \frac{3\pi}{8})) = (3\cdot 12)(\cos(\frac{\pi}{8}+\frac{3\pi}{8}) + i\sin(\frac{\pi}{8}+\frac{3\pi}{8}))$$

Step 2: Simplify the multiplication

Simplify the multiplication to get the final polar form: $$(36)(\cos \frac{4\pi}{8} + i\sin \frac{4\pi}{8}) = 36\left(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2}\right)$$ Now, we have our answer in polar form.

Step 3: Convert the polar form to rectangular form

To convert the polar form to rectangular form, we need to use the formulas: - \(a = r\cos\theta\) - \(b = r\sin\theta\) With \(r = 36\), and \(\theta = \frac{\pi}{2}\), let's apply the formulas to find the real and imaginary parts of the complex number. Real part (a): $$a = 36\cos \frac{\pi}{2} = 36 \cdot 0 = 0$$ Imaginary part (b): $$b = 36\sin \frac{\pi}{2} = 36 \cdot 1 = 36$$ So our answer in rectangular form is: $$0 + 36i$$ Now we have our answer in both polar and rectangular forms: $$36\left(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2}\right) = 0 + 36i$$

Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number is a way of expressing the number in terms of its magnitude and its angle relative to the positive x-axis. This form is particularly useful when dealing with multiplication or division of complex numbers. In polar form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (also called the modulus) and \( \theta \) is the angle (also called the argument). Visually, if you imagine the complex number as a point on the complex plane, \( r \) is the distance from the origin to the point, and \( \theta \) is the angle formed with the positive x-axis.

To multiply complex numbers in polar form, you multiply their magnitudes and add their angles, as demonstrated in the textbook exercise solution. This operation is akin to rotating and stretching the position of a point on the complex plane, which is why multiplication in polar form is so intuitive.

Rectangular Form of Complex Numbers

When we speak of the rectangular form of complex numbers, we are referring to the standard algebraic notation \( a + bi \), where \( a \) represents the real part and \( b \) represents the imaginary part. It is termed 'rectangular' because the components define the sides of a rectangle in the complex plane, with \( a \) along the horizontal axis (real axis) and \( b \) along the vertical axis (imaginary axis).

The rectangular form is the form most people are familiar with and is most useful for addition and subtraction of complex numbers. This is because these operations can be performed by simply adding or subtracting the corresponding real and imaginary parts of the complex numbers. The step-by-step solution provides an example of converting from polar to rectangular form, which is valuable for students who need to present their answer in this common format.

Converting Polar to Rectangular Form

The conversion from polar to rectangular form is a process that allows us to switch between the two forms of a complex number. This is achieved through the use of trigonometric identities, specifically the cosine and sine of the angle \( \theta \). The formulas involved are quite straightforward:\

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• For the real part \( a \): \( a = r\cos\theta \)
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• For the imaginary part \( b \): \( b = r\sin\theta \)
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\By using these formulas, you can take a complex number expressed with a magnitude and an angle (in polar form) and calculate the equivalent real and imaginary parts for the rectangular form. This is precisely what was shown in the final step of the solution for the exercise. It's important to remember that these conversions rely on an understanding of the unit circle and the behavior of sine and cosine values at different angles, which is critical for students learning about complex numbers.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers is another way to mention the polar form. It involves expressing a complex number using a trigonometric representation involving sine (\( \sin \)) and cosine (\( \cos \)). It mirrors the polar form exactly, with the notation \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument of the complex number. This form is especially powerful when dealing with problems that involve rotation or periodicity, where trigonometric functions naturally occur.

In the context of our textbook problem, by using the trigonometric (polar) form, we can swiftly carry out the multiplication of the given complex numbers. This utilization highlights the elegance and simplicity of using the trigonometric form for such operations in complex analysis.