Problem 25 - Precalculus

Question

Use a calculator to express each complex number in polar form. $$3-7 i$$

Step-by-Step Solution

Answer

The polar form is \(\sqrt{58}(\cos(-1.1659) + i\sin(-1.1659))\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(3 - 7i\). Here, the real part is 3 and the imaginary part is -7.

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). For \(3 - 7i\), calculate:\[r = \sqrt{3^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58}\]

Step 3: Find the Argument

The argument \(\theta\) of a complex number is found using the arctangent function: \(\theta = \tan^{-1} \left( \frac{b}{a} \right)\). For \(3 - 7i\), calculate:\[\theta = \tan^{-1} \left( \frac{-7}{3} \right)\]Using a calculator, \(\theta\) is approximately \(-1.1659\) radians.

Step 4: Express in Polar Form

The polar form of a complex number is given by \(r(\cos(\theta) + i\sin(\theta))\). Substitute the magnitude and argument:\[\sqrt{58}(\cos(-1.1659) + i\sin(-1.1659))\]

Key Concepts

Magnitude of Complex Number

In the realm of complex numbers, understanding how to determine the magnitude is essential. Each complex number is made up of a real and an imaginary part, often expressed as \(a + bi\). The magnitude, or otherwise known as the modulus, can be thought of as the length of the vector from the origin to the point \((a, b)\) on the complex plane.
To find this magnitude, you can rely on the Pythagorean Theorem. The magnitude \(r\) is calculated using:

• Formula: \( r = \sqrt{a^2 + b^2} \)
• Example: Let's take the complex number \(3 - 7i\), where \(a = 3\) and \(b = -7\).

Compute the magnitude as follows: \( r = \sqrt{3^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58} \).
The result, \(\sqrt{58}\), represents the distance from the origin to the point \(3 - 7i\) in the complex plane.

Argument of Complex Number

The argument of a complex number is the angle formed in the complex plane between the positive real axis and the line representing the complex number. It's a crucial aspect in the conversion to polar form.
To calculate the argument \(\theta\):

• Use: \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \)
• Direction: The angle is measured counterclockwise from the positive real axis.

For the complex number \(3 - 7i\), substitute \(a = 3\) and \(b = -7\) into the formula:

• Calculate: \( \theta = \tan^{-1} \left( \frac{-7}{3} \right) \)

Use a calculator to find \(\theta\) which is approximately \(-1.1659\) radians. This negative value indicates the angle is measured clockwise from the positive axis, a typical scenario when the imaginary part \(b\) is negative.

Complex Numbers

Complex numbers extend the idea of numbers into a two-dimensional plane, providing a comprehensive system for representing numbers. In the form \(a + bi\), \(a\) is the real part and \(bi\) is the imaginary part, where \(i\) is the imaginary unit defined by the property \(i^2 = -1\).
This truly unique structure allows complex numbers to handle problems that are not easily solved in the realm of real numbers alone.

• Representation: Any complex number can be graphically represented by a point on the complex plane, having horizontal (real) and vertical (imaginary) components.
• Operations: You can perform addition, subtraction, multiplication, and division with complex numbers, much like real numbers.

When transforming a complex number into polar form, both the magnitude and the argument come together in concise notation: \(r (\cos(\theta) + i\sin(\theta))\). This succinctly embodies both the distance and direction within the plane, offering a full picture of the complex number's qualities."