Complex Analysis - Maths KU

4.3 Trigonometric and Hyperbolic Functions 201
It follows from (2) and (3) that the complex sine and cosine functions defined
by (4) agree with the real sine and cosine functions for real input. Analogous to
real trigonometric functions, we next define the complex tangent, cotangent,
secant, and cosecant functions using the complex sine and cosine:
tan z =
sin z
cos z
1
1
, cot z = , sec z = , and csc z = . (5)
cos z sin z cos z sin z
These functions also agree with their real counterparts for real input.
EXAMPLE 1 Values of Complex Trigonometric Functions
In each part, express the value of the given trigonometric function in the form
a + ib.
(a) cos i (b) sin (2 + i) (c) tan (π − 2i)
Solution For each expression we apply the appropriate formula from (4) or
(5) and simplify.
(a) By (4),
(b) By (4),
cos i = ei·i + e −i·i
2
= e−1 + e
2
≈ 1.5431.
sin (2 + i) = ei(2+i) − e−i(2+i) 2i
= e−1+2i − e1−2i 2i
= e−1 (cos2 + i sin 2) − e(cos(−2) + i sin(−2))
0.9781 + 2.8062i
≈
2i
≈ 1.4031 − 0.4891i.
(c) By the first entry in (5) together with (4) we have:
�
i(π−2i) −i(π−2i) e − e
tan (π − 2i) =
�� 2i
�
ei(π−2i) + e−i(π−2i) �� 2 = ei(π−2i) − e−i(π−2i) �
ei(π−2i) + e−i(π−2i) � i
= − e2 − e−2 e2 i ≈−0.9640i.
+ e−2 Identities Most of the familiar identities for real trigonometric functions
hold for the complex trigonometric functions. This follows from Definition 4.6
and properties of the complex exponential function. We now list some of the
2i