Based on the results obtained in the method for defining the complex cosine, which of the following is the definition of the complex sine function ?
We define and discuss the complex trigonometric functions.
The Complex Cosine
To define we will use Maclaurin series and the sum identity for the cosine.
The series of interest are:
and the sum identity for the cosine is: We get the ball rolling by allowing an imaginary term in the sum identity: Next, we define the sine and cosine of a purely imaginary angle using their respective power series: and These power series can be simplified into hyperbolic functions (!) by noting that for all : and
Substituting these into the sum identity establishes We use this as our definition of the complex cosine.
The Complex Sine
The sum identity for the sine function states that for all angles and .
example 1 Solve for .
Since we need simultaneously. The second equation gives We proceed on a case by case basis. If , then since we would be led to which has no solutions.
Next, if where is even, then and we arrive at which has two solutions (verify) Finally, if where is odd, then and we arrive at which has no solutions (verify). Thus the equation has as its solutions where is any integer.
Since we need simultaneously. The second equation gives We proceed on a case by case basis. If , then since we would be led to which has no solutions.
Next, if where is even, then and we arrive at which has two solutions (verify) Finally, if where is odd, then and we arrive at which has no solutions (verify). Thus the equation has as its solutions where is any integer.
Identities
Periodicity
Since the real sine and cosine functions are periodic, so are their complex extenstions.
The periodicity follows immediately from the definition: The periodicity of the cosine is proved similarly (verify).Evenness and Oddness
Recall that and are odd functions and that and are even functions. As a result, is an odd function and is an even function.
To obtain the first equation, we have The evenness of the complex cosine is demonstrated similarly (verify).Shift by
The proof of these equations follows directly from their real counterparts (verify).Relation to the Complex Exponential
To prove the first equation, we begin with and Subtracting the second of these from the first, we get The result follows by dividing by . The proof of the second equation is similar (verify).Co-functions
These identities follow directly from their counterparts for the real sine and cosine functions.Sum and Difference Identities
To prove the first equation, we rewrite the right hand side using the complex exponential. The first term is The second term is Adding these, we getThe second equation follows from the first by replacing with and using evenness and oddness. The third and fourth equations are proved in the same manner as the first and second (verify).
The Other Trig functions
The other four trigonometric functions are defined in terms of the sine and cosine.
The functions and are -periodic and the functions and are -periodic (verify). The Pythagorean Identity for the sine and cosine gives rise to two other Pythagorean identities: and