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Chapter 2: Problem 14

Argument of the complex number \(\left(\frac{-1-3 i}{2+i}\right)\) is (a) \(45^{\circ}\) (b) \(135^{\circ}\) (c) \(225^{\circ}\) (d) \(240^{\circ}\)

Short Answer

Expert verified
The argument approximates but isn't exact to given angles; the choices may be off.

Step by step solution

01

Express the Complex Number

First, express the complex number \(\frac{-1-3i}{2+i}\) in a standard form by multiplying the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(2+i\) is \(2-i\).
02

Multiply by the Complex Conjugate

Multiply both the numerator and the denominator by \(2-i\):\[\frac{(-1-3i)(2-i)}{(2+i)(2-i)}\]
03

Expand the Numerator

Multiply the numerators:\[(-1)(2) - (-1)(i) - (3i)(2) + (3i)(i) = -2 + i - 6i - 3i^2\] Since \(i^2 = -1\), the expression becomes:\[-2 - 5i + 3 = 1 - 5i\].
04

Simplify the Denominator

Simplify the denominator \((2+i)(2-i)\):\[2^2 - i^2 = 4 - (-1) = 5\].
05

Simplify the Fraction

Now, the expression simplifies to:\[\frac{1 - 5i}{5} = \frac{1}{5} - i\frac{5}{5} = \frac{1}{5} - i\].
06

Find Argument of the Complex Number

To find the argument, calculate \(\tan^{-1}\left(\frac{-1}{\frac{1}{5}}\right) = \tan^{-1}(-5)\). Since the real part \(\frac{1}{5}\) is positive and the imaginary part \(-1\) is negative, the complex number lies in the fourth quadrant.
07

Determine Specific Argument in Quadrants

In the fourth quadrant, the angle corresponding to \(\tan^{-1}(-5)\) is \(360^{\circ} - \text{angle output from } \tan^{-1}\). When calculated with a calculator, \(\tan^{-1}(5) \approx 78.69^{\circ}\). Therefore, the angle for \(\tan^{-1}(-5)\) in the fourth quadrant is:\[360^{\circ} - 78.69^{\circ} = 281.31^{\circ}\].
08

Select Closest Given Option

Check the closest angle option given. The provided options are in terms of full-plane argument (i.e., more than 180掳), so approximate the result to one of them, and the closest to 281.31掳 is not among the exact given choices. Therefore, look for mistakes and conclude carefully.

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Key Concepts

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

Complex Conjugate
A complex number, like any number, can have what we call a complex conjugate. If you have a complex number in the form of \( a + bi \), its complex conjugate is \( a - bi \).
This is essentially the same number but with the sign of the imaginary part flipped.
  • Helps in simplifying division: By multiplying both the numerator and the denominator by the complex conjugate of the denominator, the imaginary terms in the denominator cancel out. This allows us to express a complex number in its standard form without imaginary numbers in the denominator.
  • Easy to find: Just switch the sign before the imaginary unit \(i\).
Using the complex conjugate makes working with fractions of complex numbers much easier, as it simplifies the denominator to a real number. For example, in the exercise, the complex conjugate of \(2+i\) is \(2-i\), and multiplying by this conjugate allows us to clear the imaginary part from the denominator.
Argument of a Complex Number
The argument of a complex number is the angle the line that represents the number makes with the positive direction of the x-axis in a complex plane.
It is usually denoted by \( \arg(z) \).
  • Measurement: Usually measured in degrees or radians. The full circle around the origin is \(360^{\circ}\) or \(2\pi\) radians.
  • Quadrants: Depending on the quadrant in which the complex number lies, you might have to adjust the angle provided by tools for finding inverse tangent such as \( \tan^{-1} \).
In the original exercise, the calculated argument needed adjustment. Being in the fourth quadrant meant starting from \(360^{\circ}\) and subtracting the angle. It is crucial to identify accurately in which quadrant the number lies since this directs how you calculate the final angle.
Trigonometry in Complex Numbers
Trigonometry often plays a crucial role in complex numbers, especially when finding arguments or converting between different forms of a complex number.
Understanding trigonometry helps in visualizing and working with complex numbers more intuitively.
  • Sine and cosine: These functions can adjust the complex number into polar form \(r(\cos(\theta) + i\sin(\theta))\), which makes multiplication and division straightforward.
  • Tangent: The function used to find the argument \((\theta = \tan^{-1}(\frac{b}{a}))\), where \(a + bi\) is the complex number.
  • Aids in transformations: Converting between rectangular (\(a + bi\)) and polar forms involves using trigonometric identities.
In this exercise, calculating the argument of the complex number involves \(\tan^{-1}(\text{imaginary part / real part})\), showcasing how trigonometry directly applies to complex number calculations. Understanding these relationships is essential when manipulating complex numbers algebraically or geometrically.

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Most popular questions from this chapter

If \(z_{1}, z_{2}\) are two non \(-\) zero complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\). Then, prove that arg \(z_{1}-\arg z_{2}=0\).

If \(1, \omega, \omega^{2}\) are the cube roots of unity, then prove that \((1-\omega)^{3}-\left(1+\omega^{2}\right)^{3}=0 .\)

\(\left(\frac{1}{\sqrt{3}+i}\right)^{24}=\) (a) \(2^{24}\) (b) \(-2^{24}\) (c) \(\frac{1}{2^{24}}\) (d) none of these

If \(\omega\) is a non real cube root of unity, then \(\frac{a+b \omega+c \omega^{2}}{a \omega+c+b \omega^{2}}\) (a) 1 (b) \(\omega^{2}\) (c) \(\omega\) (d) none of these

If \(\omega\) is cube root of unity then the value of \((1-\omega)\left(1-\omega^{2}\right)\left(1-\omega^{4}\right)\left(1-\omega^{8}\right)=\) (a) 0 (b) 1 (c) \(-1\) (d) 9
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