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This module is part of the collection, A First Course in Electrical and Computer Engineering . The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

Let's try to extend our definitions of the function e x to the argument x = j Θ . Then e j Θ is the function

e j θ = lim n ( 1 + j θ n ) n

The complex number 1 + j θ n is illustrated in [link] . The radius to the point 1 + j θ n is r = ( 1 + θ 2 n 2 ) 1 / 2 and the angle is φ = tan - 1 θ n This means that the n t h power of 1 + j θ n has radius r n = ( 1 + θ 2 n 2 ) n / 2 and angle n φ = n tan - 1 θ n (Recall our study of powers of z . ) Therefore the complex number ( 1 + j θ n ) n may be written as

( 1 + j θ n ) n = ( 1 + θ 2 n 2 ) n / 2 [ cos ( n tan - 1 θ n ) + j sin ( n tan - 1 θ n ) ] .

For n large, ( 1 + θ 2 n 2 ) n / 2 2 1 , and n tan - 1 θ n n θ n = θ . Therefore ( 1 + j θ n ) n is approximately

( 1 + j θ n ) n = 1 ( cos θ + j sin θ ) .

( 1 + j θ n ) n = 1 ( cos θ + j sin θ ).

This finding is consistent with our previous definition of e j θ !

Figure one shows a cartesian graph with a labeled line segment beginning at the origin and extending into the first quadrant. The location of the final point of the segment is labeled with a horizontal value, 1, and a vertical value, θ/n. At the end of the line segment, an expression reads, 1 + j * θ/n. Figure one shows a cartesian graph with a labeled line segment beginning at the origin and extending into the first quadrant. The location of the final point of the segment is labeled with a horizontal value, 1, and a vertical value, θ/n. At the end of the line segment, an expression reads, 1 + j * θ/n.
The Complex Number 1 + j θ n

The series expansion for e j θ is obtained by evaluating Taylor's formula at x = j θ :

e j θ = n = 0 1 n ! ( j θ ) n .

When this series expansion for e j θ is written out, we have the formula

e j θ = n = 0 1 ( 2 n ) ! ( j θ ) 2 n + n = 0 1 ( 2 n + 1 ) ! ( j θ ) 2 n + 1 = n = 0 ( - 1 ) n ( 2 n ) ! θ 2 n + j n = 0 ( - 1 ) n ( 2 n + 1 ) ! θ 2 n + 1 .

It is now clear that cos θ and sin θ have the series expansions

cos θ = n = 0 ( - 1 ) n ( 2 n ) ! θ 2 n
sin θ = n = 0 ( - 1 ) n ( 2 n + 1 ) ! θ 2 n + 1 .

When these infinite sums are truncated at N - 1 , then we say that we have N-term approximations for cos θ and sin θ :

cos θ n = 0 N - 1 ( - 1 ) n ( 2 n ) ! θ 2 n
sin θ n = 0 N - 1 ( - 1 ) n ( 2 n + 1 ) ! θ 2 n + 1 .

The ten-term approximations to cos θ and sin θ are plotted over exact expressions for cos θ and sin θ in [link] . The approximations are very good over one period ( 0 θ 2 π ) , but they diverge outside this interval. For more accurate approximations over a larger range of θ ' s , we would need to use more terms. Or, better yet, we could use the fact that cos θ and sin θ are periodic in θ . Then we could subtract as many multiples of 2 π as we needed from θ to bring the result into the range [ 0 , 2 π ] and use the ten-term approximations on this new variable. The new variable is called θ -modulo 2 π .

Figure two shows a graph with multiple winding lines resembling sinusoidal patterns. The horizontal axis is labeled, angle in radians, and its values range from 0 to 14 in increment of 2. The vertical axis is labeled, sine and cosine, and its values range from -1 to 1 in increments of 0.5. The lines on the graph will be read from left to right. At horizontal point zero, two curves begin, one following closely to a sine curve, beginning at vertical value 0 and increasing to 1, before decreasing to -1 and repeating, and the other following closely to a cosine curve, beginning at 1 and decreasing to -1, then returning to increase to 1 and repeating. These two curves repeat in their predictable pattern to approximately horizontal value 12. At approximately the horizontal value 8, two curves branch off of the more predictable curves and resemble distorted sinusoidal directions. Off of the sine curve at a vertical value of 1 is a curve that begins downward but at a more shallow slope than the sine curve, then increases sharply to terminate at approximately (9.5, 1). Off of the cosine curve at a vertical value of 0, when the cosine curve is moving in a negative direction, the spinoff curve begins downward but at a shallower slope than the cosine curve, and then it sharply begins increasing to its termination point of approximately (9.25, 1) Figure two shows a graph with multiple winding lines resembling sinusoidal patterns. The horizontal axis is labeled, angle in radians, and its values range from 0 to 14 in increment of 2. The vertical axis is labeled, sine and cosine, and its values range from -1 to 1 in increments of 0.5. The lines on the graph will be read from left to right. At horizontal point zero, two curves begin, one following closely to a sine curve, beginning at vertical value 0 and increasing to 1, before decreasing to -1 and repeating, and the other following closely to a cosine curve, beginning at 1 and decreasing to -1, then returning to increase to 1 and repeating. These two curves repeat in their predictable pattern to approximately horizontal value 12. At approximately the horizontal value 8, two curves branch off of the more predictable curves and resemble distorted sinusoidal directions. Off of the sine curve at a vertical value of 1 is a curve that begins downward but at a more shallow slope than the sine curve, then increases sharply to terminate at approximately (9.5, 1). Off of the cosine curve at a vertical value of 0, when the cosine curve is moving in a negative direction, the spinoff curve begins downward but at a shallower slope than the cosine curve, and then it sharply begins increasing to its termination point of approximately (9.25, 1)
Ten-Term Approximations to cos θ and sin θ

Demo 2.1 (MATLAB). Create a MATLAB file containing the following demo MATLAB program that computes and plots two cycles of cos θ and sin θ versus θ . You should observe [link] . Note that two cycles take in 2 ( 2 π ) radians, which is approximately 12 radians.

Figure three shows a graph with multiple winding lines resembling sinusoidal patterns. The horizontal axis is labeled, angle in radians, and its values range from 0 to 14 in increment of 2. The vertical axis is labeled, sine and cosine, and its values range from -1 to 1 in increments of 0.5. The lines on the graph will be read from left to right. At horizontal point zero, two curves begin, one following closely to a sine curve, beginning at vertical value 0 and increasing to 1, before decreasing to -1 and repeating, and the other following closely to a cosine curve, beginning at 1 and decreasing to -1, then returning to increase to 1 and repeating. These two curves repeat in their predictable pattern to approximately horizontal value 12. Figure three shows a graph with multiple winding lines resembling sinusoidal patterns. The horizontal axis is labeled, angle in radians, and its values range from 0 to 14 in increment of 2. The vertical axis is labeled, sine and cosine, and its values range from -1 to 1 in increments of 0.5. The lines on the graph will be read from left to right. At horizontal point zero, two curves begin, one following closely to a sine curve, beginning at vertical value 0 and increasing to 1, before decreasing to -1 and repeating, and the other following closely to a cosine curve, beginning at 1 and decreasing to -1, then returning to increase to 1 and repeating. These two curves repeat in their predictable pattern to approximately horizontal value 12.
The Functions cos θ and sin θ

The Unit Circle. The unit circle is defined to be the set of all complex numbers z whose magnitudes are 1. This means that all the numbers on the unit circle may be written as z = e j θ . We say that the unit circle consists of all numbers generated by the function z = e j θ as θ varies from 0 to 2 π . See [link] .

A Fundamental Symmetry. Let's consider the two complex numbers z 1 and 1 z 1 * , illustrated in [link] . We call 1 z 1 * t h e “reflection of z through the unit circle” (and vice versa). Note that z 1 = r 1 e j θ 1 and 1 z 1 * = 1 r 1 e j θ 1 . The complex numbers z 1 - e j θ and 1 z 1 * - e j θ are illustrated in [link] . The magnitude squared of each is

| z 1 - e j θ | 2 = ( z 1 - e j θ ) ( z 1 * - e - j θ )
| 1 z 1 * - e j θ | 2 = ( 1 z 1 * - e j θ ) ( 1 z 1 - e - j θ ) .

The ratio of these magnitudes squared is

β 2 = ( z 1 - e j θ ) ( z 1 * - e - j θ ) ( 1 z 1 * - e j θ ) ( 1 z 1 - e - j θ )

This ratio may be manipulated to show that it is independent of θ , meaning that the points z 1 and 1 z 1 * maintain a constant relative distance from every point on the unit circle:

β 2 = e j θ ( e - j θ z 1 - 1 ) ( z 1 * e j θ - 1 ) e - j θ 1 z i ( 1 - e j θ z 1 * ) ( 1 - z 1 e - j θ ) 1 z 1 = | z 1 | 2 , independent of θ !

This result will be of paramount importance to you when you study digital filtering, antenna design, and communication theory.

Figure four is a cartesian graph with a unit circle and multiple labeled points connected by line segments. A point on the circle labeled e^(jθ) is located in quadrant one. A point inside the circle in quadrant two is labeled z_1. A point located outside the circle in the second quadrant is labeled 1/z_1^*. All points are connected to each other and to the origin with unlabeled line segments. An arc measures the angle between the line segment connecting the origin to e^(jθ), and is labeled θ. Figure four is a cartesian graph with a unit circle and multiple labeled points connected by line segments. A point on the circle labeled e^(jθ) is located in quadrant one. A point inside the circle in quadrant two is labeled z_1. A point located outside the circle in the second quadrant is labeled 1/z_1^*. All points are connected to each other and to the origin with unlabeled line segments. An arc measures the angle between the line segment connecting the origin to e^(jθ), and is labeled θ.
The Unit Circle

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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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