Explanation Given information: − 4 + 4 i a . 4 2 ( cos 135 ° + i sin 135 ° ) b . 4 ( cos 315 ° + i sin 315 ° ) c . 4 2 ( cos 315 ° + i sin 315 ° ) d . 4 2 ( sin 135 ° + i cos 135 ° ) Concept Involved: To graphically represent a complex number a + i b , we need a system which allows us to indicate the values of both ‘a’ and ‘b’. To do this, we set up a rectangular coordinate system much like the Cartesian coordinate system, except that we use the horizontal axis to indicate the real part and the vertical axis to indicate the imaginary part. The resulting two-dimensional coordinate system is called the complex plane, or Argand plane. Graph of a complex number: The graph of the complex number x + y i is a vector (arrow) that extends from the origin out to the point ( x , y ) in the complex plane. Modulus: The absolute value or modulus of the complex number z = x + y i is the distance from the origin to the point ( x , y ) in the complex plane. If this distance is denoted by r , then r = | z | = | x + y i | = x 2 + y 2 Argument: The argument of the complex number z = x + y i , denoted a r g ( z ) , is the smallest positive angle θ from the positive real axis to the graph of z. Trigonometric Form: If z = x + y i is a complex number in standard form, then the trigonometric form of z is given by: z = r ( cos θ + i sin θ ) = r c i s θ Where r is the modulus of z and ? is the argument of z. We can convert back and forth between standard form and trigonometric form using the relationships that follow: For z = x + y i = r ( cos θ + i sin θ ) = r c i s θ r = x 2 + y 2 and θ is such that cos θ = x r , sin θ = y r , a n d tan θ = y x The sign of trigonometric ratio in different quadrants

8th Edition

ISBN: 9781305652224

Author: Charles P. McKeague, Mark D. Turner

Publisher: Cengage Learning

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 8.2, Problem 90PS

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Given complex number in trigonometric form.

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Chapter 8.2, Problem 89PS

Chapter 8.2, Problem 91PS

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Given complex number in trigonometric form.

Solution Summary: The author explains how to graphically represent a complex number in trigonometric form.

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Chapter 8 Solutions

To write:

Given complex number in trigonometric form.