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Chapter 31: Problem 986

Express each of the following in trignometric form (c) \(2+i\)

Short Answer

Expert verified
The complex number \(2+i\) in trigonometric form is: \[\sqrt{5}(\cos{\tan^{-1}{\frac{1}{2}}} + i\sin{\tan^{-1}{\frac{1}{2}}})\]

Step by step solution

01

Calculate the magnitude of the complex number

We can calculate the magnitude (or modulus) of a complex number \(a+bi\) using the formula: \[r = \sqrt{a^2 + b^2}\] In this case, \(a=2\) and \(b=1\). So, \[r = \sqrt{2^2 + 1^2} = \sqrt{5}\]
02

Calculate the argument of the complex number

To find the argument (or angle) of \(2+i\), we can use the following formula: \[\theta = \tan^{-1}{\frac{b}{a}}\] Here, \(a=2\) and \(b=1\). So, we have: \[\theta = \tan^{-1}{\frac{1}{2}}\]
03

Express the complex number in trigonometric form

Now we have the magnitude \(r = \sqrt{5}\) and the argument \(\theta = \tan^{-1}{\frac{1}{2}}\). We can write the complex number in trigonometric form as: \[z = r(\cos{\theta} + i\sin{\theta})\] Therefore, the complex number \(2+i\) in trigonometric form is: \[2+i = \sqrt{5}(\cos{\tan^{-1}{\frac{1}{2}}} + i\sin{\tan^{-1}{\frac{1}{2}}})\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnitude of Complex Numbers
The magnitude, or modulus, of a complex number is a measure of its size or length in the complex plane. Imagine the complex number as a point with coordinates. The magnitude tells us how far that point is from the origin. For a complex number expressed as \(a + bi\), the magnitude \(r\) is calculated using the Pythagorean theorem:
  • \[ r = \sqrt{a^2 + b^2} \]
For example, take the complex number \(2 + i\). Here, \(a = 2\) and \(b = 1\). Substituting these values into our formula gives:
  • \[ r = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]
Thus, the magnitude of \(2 + i\) is \(\sqrt{5}\). This tells us that the "distance" of the point from the origin in the complex plane is \(\sqrt{5}\) units.
Argument of Complex Numbers
The argument of a complex number provides the direction of the number relative to the positive real axis. It's essentially the angle that the line representing the complex number makes with the real axis. This argument is denoted by \(\theta\) and calculated using the inverse tangent function:
  • \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
Considering the complex number \(2 + i\), we have \(a = 2\) and \(b = 1\). Thus, the argument is:
  • \[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]
It's important to note that while the result from \(\tan^{-1}\) typically gives an acute angle, the quadrant of the complex number should also be considered to determine the true direction. For \(2 + i\), since both terms are positive, it lies in the first quadrant.
Expressing Complex Numbers in Polar Form
The trigonometric or polar form of a complex number beautifully combines its magnitude and argument into a single statement. This form is particularly useful for multiplicative operations on complex numbers. In polar form, any complex number \(z = a + bi\) is expressed as:
  • \[ z = r(\cos \theta + i \sin \theta) \]
This utilizes the magnitude \(r\) and argument \(\theta\) we calculated earlier. For the complex number \(2 + i\), we found \(r = \sqrt{5}\) and \(\theta = \tan^{-1}\left(\frac{1}{2}\right)\).
  • Therefore, its polar form is:
  • \[ 2 + i = \sqrt{5}(\cos({\tan^{-1}\left(\frac{1}{2}\right)}) + i\sin({\tan^{-1}\left(\frac{1}{2}\right)})) \]
This representation effectively provides a way to perform operations like multiplication and division much more simply than with the standard rectangular form. It is a powerful tool in both pure and applied mathematics.

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