Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\left[3\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\right]\left[2\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]$$

Step-by-Step Solution

Answer

The product in rectangular form is \( -3\sqrt{3} + 3i \).

Step 1: Convert Trigonometric Form to Rectangular Form

In trigonometric form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \), which can be converted to rectangular form as \( a + bi \), where \( a = r \cos \theta \) and \( b = r \sin \theta \). For the first term: \( 3(\cos 60^{\circ} + i \sin 60^{\circ}) = 3(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = \frac{3}{2} + i\frac{3\sqrt{3}}{2} \).For the second term:\( 2(\cos 90^{\circ} + i \sin 90^{\circ}) = 2(0 + i\cdot1) = 0 + 2i \).

Step 2: Multiply the Rectangular Forms

Now that we have both terms in rectangular form, we multiply them as follows:Multiply \( (\frac{3}{2} + i\frac{3\sqrt{3}}{2}) \) by \( (0 + 2i) \).Use the distributive property:\[(\frac{3}{2} + i\frac{3\sqrt{3}}{2})(0 + 2i) = \frac{3}{2}\cdot0 + \frac{3}{2}\cdot2i + i\frac{3\sqrt{3}}{2}\cdot0 + i\frac{3\sqrt{3}}{2}\cdot2i \].This simplifies to:\[0 + 3i + 0 - 3\sqrt{3} \].

Step 3: Combine and Simplify Terms

Combine the terms resulting from the product:Rearrange the terms: \[ 3i - 3\sqrt{3} \].So, the product in rectangular form is \[ -3\sqrt{3} + 3i \].

Key Concepts

Rectangular Form

Complex numbers can be a bit tricky at first, but once you grasp their representation, they become easier to handle. One way to represent complex numbers is the rectangular form. This form expresses complex numbers as a sum of a real part and an imaginary part, written as \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary component with \( i \) denoting the imaginary unit (\( i^2 = -1 \)).

• Real Part (\( a \)): This is the horizontal component on the complex plane.
• Imaginary Part (\( b \)): This is the vertical component on the complex plane, and it contains the imaginary unit \( i \).

For example, if we have a complex number in trigonometric form like \( 3(\cos 60^{\circ} + i \sin 60^{\circ}) \), we can convert it to rectangular form. The process involves calculating the cosine and sine values, multiplying them by the modulus, \( r \), and substituting into \( a + bi \): \[3 \times \cos 60^{\circ} + i \times 3 \times \sin 60^{\circ} = \frac{3}{2} + i \frac{3\sqrt{3}}{2}\]This gives us the rectangular form for that complex number.

Trigonometric Form

When dealing with complex numbers, the trigonometric form is also quite helpful. This form represents complex numbers in terms of magnitude and angle. It is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus (or magnitude) of the complex number and \( \theta \) is the angle measured counterclockwise from the positive x-axis in the complex plane.

• Modulus (\( r \)): Represents the distance from the origin to the point on the complex plane.
• Argument (\( \theta \)): The angle made with the positive x-axis.

For example, given the complex number \( 2(\cos 90^{\circ} + i \sin 90^{\circ}) \), the modulus here is 2. At \( 90^{\circ} \), \( \cos 90^{\circ} \) equals 0 and \( \sin 90^{\circ} \) equals 1, transforming our trigonometric form to \( 0 + 2i \) in rectangular form. This shows how the angle \( \theta \) influences the real and imaginary parts.

Complex Multiplication

Multiplying complex numbers is similar to multiplying binomials. When you have two complex numbers in rectangular form, such as \( (a + bi)(c + di) \), you use the distributive property, which involves multiplying each part of the first complex number by each part of the second.

• Multiply the real parts: \( ac \)
• Multiply the imaginary parts: \( bdi^2 \) (remember, \( i^2 = -1 \))
• Cross multiplication of real and imaginary parts: \( adi + bci \)

For instance, to multiply \( (\frac{3}{2} + i\frac{3\sqrt{3}}{2}) \) and \( (0 + 2i) \), distribute each term:
\[\left(\frac{3}{2} + i\frac{3\sqrt{3}}{2}\right)(0 + 2i) = \frac{3}{2} \cdot 0 + \frac{3}{2} \cdot 2i + i\frac{3\sqrt{3}}{2} \cdot 0 + i\frac{3\sqrt{3}}{2} \cdot 2i\]Simplify the multiplication and combine like terms to get the result \(-3\sqrt{3} + 3i\). Understanding these operations makes it easier to work with complex numbers, facilitating tasks like powers and roots of complex numbers as well.