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Chapter 19: Problem 5

zis a complex number with \(|z|=1\) and \(\arg (2)=3 \pi / 4\). The value of \(z\) is (a) \(0+i y \sqrt{2}\) (b) \((-1+i) \sqrt{2}\) (c) \((1-i) \sqrt{2}\) (d) \((-1-i) \sqrt{2}\).

Short Answer

Expert verified
The correct option is (b) \((-1+i) \sqrt{2}\).

Step by step solution

01

Understand the Given Information

We are given that \(z\) is a complex number with a modulus \(|z| = 1\) and its argument \(\arg(z) = \frac{3\pi}{4}\).
02

Recall the Standard Form of a Complex Number

A complex number \(z\) with modulus \(r\) and argument \(\theta\) can be expressed in the form \(z = r(\cos \theta + i\sin \theta)\).
03

Apply the Given Conditions to the Formula

Since \(|z| = 1\) and \(\arg(z) = \frac{3\pi}{4}\), we can write:\[z = \cos\left(\frac{3\pi}{4}\right) + i \sin\left(\frac{3\pi}{4}\right).\]
04

Calculate the Trigonometric Functions

Calculate \(\cos\left(\frac{3\pi}{4}\right)\) and \(\sin\left(\frac{3\pi}{4}\right)\):- \(\cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}\)- \(\sin\left(\frac{3\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
05

Substitute Values into the Expression

Substituting these values into \(z = \cos(\theta) + i\sin(\theta)\):\[z = -\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} = \left(-1 + i\right)\frac{1}{\sqrt{2}}.\]
06

Simplify and Identify the Correct Option

Multiply the numerator and the denominator by \(\sqrt{2}\) to match the options:\[z = \left(-1 + i\right)\sqrt{2}.\]Comparing with the options, this matches option (b).

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Key Concepts

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

Modulus
The modulus of a complex number is like the length of a line segment that starts at the origin and ends at the point representing the complex number in the complex plane. For any complex number expressed as \(z = a + bi\), its modulus is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). This is analogous to finding the hypotenuse of a right triangle.
  • In our problem, the modulus is given as \(|z| = 1\). This means the point representing \(z\) lies on a unit circle centered at the origin in the complex plane.
  • The unit circle aspect signifies that the distance of the complex number from the origin is exactly 1, simplifying many calculations such as deriving the angle or argument.
Since the modulus is 1, this simplifies the trigonometric form to only consider the unit values for cosine and sine functions.
Argument of a Complex Number
The argument of a complex number indicates its angle with the positive direction of the real axis. For a complex number \(z = a + bi\), the argument \( \arg(z) \) is given by \( \theta \), where \(\tan(\theta) = \frac{b}{a}\).
  • The range of the argument is typically from \(-\pi\) to \(\pi\), allowing it to express angles that cover all quadrants of the complex plane.
  • In the exercise, the argument is \(\frac{3\pi}{4}\), placing the point in the second quadrant.
For the complex number revolving around the unit circle, this argument of \(\frac{3\pi}{4}\) corresponds to angles where cosine and sine values reflect crossing the imaginary axis from the left side of the plane.
Trigonometric Form
A complex number can be expressed in trigonometric form, highlighting its rotation and magnitude in the complex plane. This is done using the equation \(z = r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus and \(\theta\) is the argument.
  • This form particularly helps in visualizing and solving problems involving rotation or multiplication of complex numbers.
  • In this exercise, the modulus \(r\) is 1, so the formula becomes simpler: \(z = \cos \left(\frac{3\pi}{4}\right) + i \sin \left(\frac{3\pi}{4}\right)\).
  • Calculating those values gives us \(-\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\), showing us that the point is aligned with standard trigonometric angles where both cosine and sine are equal in magnitude but of opposite signs.
Simplifying this expression clarifies the correct properties of the given problem and matches one of the provided options after multiplying by \(\sqrt{2}\).

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