1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.4 • TR!GONOMETRJC AND HYPERBOLIC FUNCTIONS 177

Definition 5 .6: Trigonometric funct ions

sin z cosz 1 1

tanz = --, cotz = -.-, secz =--, and cscz = --.

cosz smz cosz sinz

The series for the complex sine and cosine agree with the real sine and
cosine when z is real, so the remaining complex trigonometric functions likewise
agree with their real counterparts. What additional properties are common? For
starters, we have

We now list several additional properties, providing proofs for some and

leaving others as exercises.

• For all complex numbers z ,

sin(-z) = -sinz,

cos(-z) = cosz, and

sin^2 z + cos^2 z = 1.

The verification that sin ( -z) = - sin z and cos ( -z) = cos z comes from

substituting - z for z in Definition 5.4. We leave verification of the identity

sin^2 z + cos^2 z = 1 as an exercise (with hints).