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Let's try to extend our definitions of the function to the argument . Then is the function
The complex number is illustrated in [link] . The radius to the point is and the angle is This means that the power of has radius and angle (Recall our study of powers of ) Therefore the complex number may be written as
For large, , and . Therefore is approximately
( sin θ ).
This finding is consistent with our previous definition of !
The series expansion for is obtained by evaluating Taylor's formula at :
When this series expansion for is written out, we have the formula
It is now clear that and have the series expansions
When these infinite sums are truncated at , then we say that we have N-term approximations for and :
The ten-term approximations to and are plotted over exact expressions for and in [link] . The approximations are very good over one period , but they diverge outside this interval. For more accurate approximations over a larger range of , we would need to use more terms. Or, better yet, we could use the fact that and are periodic in . Then we could subtract as many multiples of as we needed from to bring the result into the range and use the ten-term approximations on this new variable. The new variable is called -modulo .
Demo 2.1 (MATLAB). Create a MATLAB file containing the following demo MATLAB program that computes and plots two cycles of and sin θ versus θ . You should observe [link] . Note that two cycles take in radians, which is approximately 12 radians.
The Unit Circle. The unit circle is defined to be the set of all complex numbers whose magnitudes are 1. This means that all the numbers on the unit circle may be written as . We say that the unit circle consists of all numbers generated by the function as varies from 0 to . See [link] .
A Fundamental Symmetry. Let's consider the two complex numbers and , illustrated in [link] . We call “reflection of through the unit circle” (and vice versa). Note that and . The complex numbers and are illustrated in [link] . The magnitude squared of each is
The ratio of these magnitudes squared is
This ratio may be manipulated to show that it is independent of , meaning that the points and maintain a constant relative distance from every point on the unit circle:
This result will be of paramount importance to you when you study digital filtering, antenna design, and communication theory.
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