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Chapter 8: Problem 98

Consider the following complex numbers, and work in order. $$ w=-1+i \quad \text { and } \quad z=-1-i $$ Find the trigonometric forms of \(w\) and \(z .\)

Short Answer

Expert verified
The trigonometric form of \( w \) is \( \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4})) \). The trigonometric form of \( z \) is \( \sqrt{2}(\cos(\frac{5\pi}{4}) + i\sin(\frac{5\pi}{4})) \).

Step by step solution

01

Find the Magnitude of w

Calculate the magnitude (modulus) of the complex number \( w = -1 + i \). Use the formula: \[ |w| = \sqrt{(-1)^2 + (1)^2} \]. Simplifying this gives \[ |w| = \sqrt{2} \].
02

Find the Argument of w

Determine the argument (angle) of the complex number \( w = -1 + i \). The argument \( \theta \) can be found using \[ \theta = \tan^{-1}\left( \frac{1}{-1} \right) \], noting that \( w \) is in the second quadrant. Therefore, \[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \].
03

Express w in Trigonometric Form

Combine the magnitude and argument to convert \( w = -1 + i \) to its trigonometric form: \[ w = |w|(\cos(\theta) + i\sin(\theta)) = \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4})) \].
04

Find the Magnitude of z

Calculate the magnitude of the complex number \( z = -1 - i \). Use the formula: \[ |z| = \sqrt{(-1)^2 + (-1)^2} \]. Simplifying this gives \[ |z| = \sqrt{2} \].
05

Find the Argument of z

Determine the argument of the complex number \( z = -1 - i \). The argument \( \theta \) can be found using \[ \theta = \tan^{-1}\left(\frac{-1}{-1}\right) \], noting that \( z \) is in the third quadrant. Therefore, \[ \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \].
06

Express z in Trigonometric Form

Combine the magnitude and argument to convert \( z = -1 - i \) to its trigonometric form: \[ z = |z|(\cos(\theta) + i\sin(\theta)) = \sqrt{2}(\cos(\frac{5\pi}{4}) + i\sin(\frac{5\pi}{4})) \].

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Key Concepts

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

Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined by the property that \(i^2 = -1\).

Examples of complex numbers include \(-1 + i\) and \(-1 - i\). These numbers can be plotted on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

This representation makes it easier to visualize and perform operations like addition, subtraction, multiplication, and division.
Magnitude of Complex Numbers
The magnitude or modulus of a complex number represents its distance from the origin on the complex plane. It is given by the formula \(|z| = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of the complex number, respectively.

For example, for the complex number \(w = -1 + i\), the magnitude is calculated as:
\[ |w| = \sqrt{(-1)^2 + (1)^2} = \sqrt{2} \] Similarly, for \(z = -1 - i\), the magnitude is also:
\[ |z| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \]
This value tells us how far the complex number is from the origin (0,0) in the complex plane.
Argument of Complex Numbers
The argument of a complex number is the angle it makes with the positive real axis, measured in radians. It can be found using the \(\tan^{-1}\) function, taking into account the signs of the real and imaginary parts to determine the correct quadrant.

For \(w = -1 + i\), the argument \(\theta\) is calculated as:
\[ \theta = \tan^{-1}\left(\frac{1}{-1}\right) = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \] (since \(w\) is in the second quadrant).
For \(z = -1 - i\), the argument \(\theta\) is:
\[ \theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \] (since \(z\) is in the third quadrant).
Trigonometric Form of Complex Numbers
The trigonometric form of a complex number expresses it using its magnitude and argument. It is written as:

\[ z = |z| (\cos(\theta) + i \sin(\theta)) \]

For \(w = -1 + i\), with a magnitude of \(\sqrt{2}\)\ and an argument of \(\frac{3\pi}{4}\):
\[ w = \sqrt{2} (\cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4})) \] Similarly, for \(z = -1 - i\), with the same magnitude and an argument of \(\frac{5\pi}{4}\):
\[ z = \sqrt{2} (\cos(\frac{5\pi}{4}) + i \sin(\frac{5\pi}{4})) \] This form is very helpful for multiplying and dividing complex numbers, as it simplifies these operations compared to the standard form \(a + bi\).

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Most popular questions from this chapter

For each pair of rectangular coordinates, ( \(a\) ) plot the point and (b) give two pairs of polar coordinates for the point, where \(0^{\circ} \leq \theta<360^{\circ} .\) $$(0,-3)$$

Find the polar coordinates of the points of intersection of the given curves for the specified interval of \(\theta\). $$r=2+\sin \theta, r=2+\cos \theta ; 0 \leq \theta<2 \pi$$

One of the three cube roots of a complex number is \(2+2 i \sqrt{3}\). Determine the rectangular form of its other two cube roots.

Solve each problem. The polar equation $$r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}$$ can be used to graph the orbits of the satellites of our sun, where \(a\) is the average distance in astronomical units from the sun and \(e\) is a constant called the eccentricity. The sun will be located at the pole. The table lists the values of \(a\) and \(e\). (a) Graph the orbits of the four closest satellites on the same polar grid. Choose a viewing window that results in a graph with nearly circular orbits. (b) Plot the orbits of Earth, Jupiter, Uranus, and Pluto on the same polar grid. How does Earth's distance from the sun compare to the others' distances from the sun? (c) Use graphing to determine whether or not Pluto is always farthest from the sun. $$\begin{array}{|c|c|c|}\hline \text { Satellite } & a & e \\\\\hline \text { Mercury } & 0.39 & 0.206 \\\\\hline \text { Venus } & 0.78 & 0.007 \\\\\hline \text { Earth } & 1.00 & 0.017 \\\\\hline \text { Mars } & 1.52 & 0.093 \\\\\hline \text { Jupiter } & 5.20 & 0.048 \\\\\hline \text { Saturn } & 9.54 & 0.056 \\\\\hline \text { Uranus } & 19.20 & 0.047 \\\\\hline \text { Neptune } & 30.10 & 0.009 \\\\\hline \text { Pluto } & 39.40 & 0.249 \\\\\hline\end{array}$$

Use a calculator to perform the indicated operations. Give answers in rectangular form, expressing real and imaginary parts to four decimal places. $$\left[4.6\left(\cos 12^{\circ}+i \sin 12^{\circ}\right)\right]\left[2.0\left(\cos 13^{\circ}+i \sin 13^{\circ}\right)\right]$$
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