Problem 35 - Contemporary Precalculus

Question

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$4(\cos 2+i \sin 2)$$

Step-by-Step Solution

Answer

Question: Express the complex number \(4(\cos 2 + i \sin 2)\) in rectangular form (a + bi). Answer: \(4\cos 2+4i\sin 2\)

Step 1: Multiply the real part by 4

Since we need to find the rectangular form of the given complex number, we'll first multiply the real part by 4, which is the coefficient in the polar form. We get \(4\cos 2\).

Step 2: Multiply the imaginary part by 4

Now, we'll multiply the imaginary part by 4, getting \(4i\sin 2\).

Step 3: Combine the results

We now combine the results from Step 1 and Step 2, which gives us the rectangular form of the complex number: \(4\cos 2 + 4i\sin 2\). Therefore, the complex number in the form \(a+b i\) is $$4\cos 2+4i\sin 2$$.

Key Concepts

Polar to Rectangular Conversion

Understanding the conversion from polar to rectangular form is crucial when working with complex numbers. Complex numbers can be represented in different formats, and switching between them allows for a more suitable approach depending on the problem at hand.

In polar form, a complex number is represented as \(r(\cos \theta + i\sin \theta)\), where \(|r|\) is the magnitude (or modulus) of the complex number and \(|\theta|\) is the angle (or argument) formed with the positive direction of the real axis. To convert this to rectangular form \(a + bi\), we simply multiply the magnitude \(|r|\) by the cosine of the angle to get the real part \(a\), and the magnitude by the sine of the angle to yield the imaginary part \(b\).

Using this method, let's look at an example:

• Given the polar form \(4(\cos 2 + i\sin 2)\), we first find the real part by evaluating \(4\cos 2\).
• Next, we find the imaginary part by calculating \(4i\sin 2\).
• Finally, we combine these two to express the number in rectangular form: \(4\cos 2 + 4i\sin 2\).

This process is utilized across various fields such as engineering, physics, and mathematics for simplifying the multiplication, division, and power calculations of complex numbers.

Complex Numbers

The world of complex numbers extends the real numbers by introducing the imaginary unit \(i\), defined as \(i^2 = -1\). A complex number typically takes the form \(a + bi\), where \(|a|\) is the real part and \(|b|\) is the imaginary part.

Complex numbers are essential in solving equations that have no real solutions. They also appear in various applications such as in oscillations, signal analysis, and fluid dynamics. Visualization of complex numbers occurs on the complex plane, with the horizontal axis representing the real part (Re) and the vertical axis representing the imaginary part (Im).

The beauty of complex numbers lies in their ability to rotate and scale vectors in the complex plane, as well as to perform arithmetical operations including addition, subtraction, multiplication, and division, which have unique geometric interpretations in this system.

Trigonometric Form of Complex Numbers

The trigonometric form, also known as the polar form, of a complex number is another way to represent complex numbers, highlighting their geometric properties. It is denoted as \(r(\cos \theta + i\sin \theta)\) or \(r \cdot cis(\theta)\), with 'cis' standing for 'cosine plus i times sine'.

Why is it useful?

Using the trigonometric form simplifies complex number operations like multiplication, division, and finding powers or roots. When multiplying two complex numbers in trigonometric form, you multiply their magnitudes and add their angles. For division, you divide magnitudes and subtract angles. This form is particularly handy when dealing with problems involving rotations in the complex plane.

Conversion to the trigonometric form from rectangular form involves finding the magnitude \(r\) and the angle \(|\theta|\) with trigonometric functions. Conversely, as seen in the exercise, conversion back to rectangular form involves using the known values of \(r\) and \(|\theta|\) to compute the real and imaginary components via the cosine and sine functions respectively.