Problem 41 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$-20$$

Step-by-Step Solution

Answer

The polar form of the complex number is \(20(\cos \pi + i \sin \pi)\).

Step 1: Identify the Complex Number

The complex number given is \(-20\). This is a real number and can be written in the form of a complex number as \(-20 + 0i\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is calculated as:\[r = \sqrt{a^2 + b^2}\]For the given complex number \(-20 + 0i\), we have:\[r = \sqrt{(-20)^2 + 0^2} = \sqrt{400} = 20\]

Step 3: Determine the Angle

To find the argument \(\theta\) of the complex number, use the formula:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]However, since \(a = -20\) and \(b = 0\), the complex number lies on the negative real axis. Therefore, the angle \(\theta\) is \(\pi\).

Step 4: Write in Polar Form

The polar form of a complex number is given by:\[r(\cos\theta + i\sin\theta)\]Substitute \(r = 20\) and \(\theta = \pi\):\[20(\cos \pi + i \sin \pi)\]

Step 5: Simplify the Expression

Using trigonometric values:- \(\cos \pi = -1\)- \(\sin \pi = 0\)Thus, the expression becomes:\[20(-1 + 0i) = -20\]

Key Concepts

Polar Form

Complex numbers can be represented in a form that links directly to their location on the complex plane. This form is known as the polar form. Instead of using real and imaginary components, polar form expresses a complex number in terms of its magnitude and argument. The polar form of a complex number is typically written as \[r(\cos\theta + i\sin\theta)\]where \(r\) is the magnitude of the complex number, and \(\theta\) is the argument.
The polar form offers a more intuitive understanding of complex numbers, especially for multiplication and division.
In our case, the number \(-20 + 0i\)is expressed in polar form as \[20(\cos\pi + i\sin\pi)\]. This not only indicates the quantity (or magnitude) of the number but also the direction (or angle) on the complex plane.

Magnitude

The magnitude of a complex number is essentially how far the number is from the origin on the complex plane. It is a measure of the number's absolute size, irrespective of direction.
For a complex number expressed as \(a + bi\),the magnitude \(r\) is determined using the formula:\[r = \sqrt{a^2 + b^2}\]This calculation is similar to finding the hypotenuse of a right triangle with sides \(a\) and \(b\).
For the complex number \(-20 + 0i\),we find the magnitude \(r\)to be \(20\).
This translates to the fact that the complex number is 20 units away from the origin.
Finding magnitude is particularly important in polar form as it sets the scale of the complex number.

Argument

The argument of a complex number provides information about the direction in which the number is located on the complex plane relative to the positive real axis. It is essentially the angle the complex number makes with this axis.
To find the argument \(\theta\), we employ the formula:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]However, it's essential to consider the signs of \(a\) and \(b\) to determine the correct quadrant for \(\theta\).

• For numbers along the positive real axis, \(\theta = 0\).
• For numbers along the negative real axis, like \(-20 + 0i\), the argument \(\theta\) is \(\pi\).

Thus, for \(-20 + 0i\), the angle is \(\pi\), meaning the number points directly along the negative real axis.