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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 11
Capitolo 5, Problema 11

In Exercises 11–14, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. 1 − i

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Identify the complex number given: \(1 - i\). Here, the real part is \(1\) and the imaginary part is \(-1\).
Plot the complex number on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The point will be at coordinates \((1, -1)\).
Calculate the modulus (or magnitude) \(r\) of the complex number using the formula \(r = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2}\). Substitute the values to get \(r = \sqrt{1^2 + (-1)^2}\).
Find the argument (or angle) \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right)\). Substitute the values to get \(\theta = \tan^{-1}\left(\frac{-1}{1}\right)\).
Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), where \(r\) is the modulus and \(\theta\) is the argument you calculated.

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Complex Numbers and the Complex Plane

A complex number is expressed as a + bi, where a is the real part and b is the imaginary part. It can be represented as a point or vector in the complex plane, with the x-axis as the real axis and the y-axis as the imaginary axis. Plotting involves locating the point (a, b) on this plane.
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Dividing Complex Numbers

Polar Form of Complex Numbers

Polar form expresses a complex number using its magnitude (distance from the origin) and argument (angle with the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r = √(a² + b²) and θ = arctangent(b/a). This form highlights the number's geometric properties.
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Complex Numbers In Polar Form

Calculating the Argument (Angle)

The argument θ of a complex number is the angle between the positive real axis and the line connecting the origin to the point (a, b). It can be found using θ = arctan(b/a), adjusted for the correct quadrant. The angle can be expressed in degrees or radians depending on the context.
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Double Angle Identities
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