Trigonometry of the complex numbers

The complex sine and cosine functions are not bounded if they are defined over the set of all complex numbers. Complex sine (cosine) functions could be equal to 2 or 5.

Let us find such complex z that Sin(z) = a where a is real and a >1.
For example Sin(z) = 2 or Sin(z) = 5.

Sin(z) = = a               (1)

2ia                             (2)

Multiply (2)

Let ,                                       (3)

then

We consider a is real and a >1. It means that and

iz = Ln

z = =

= =

=

Ln(i)= i( 1+2k), k = 0, ±1, ±2, ...

z =

So, when a = 2

z =

= 1.7320;
Ln(3.7320) = 1.3169;
Ln(0.2680) = -1.3169

Thus the equation Sin(z) = 2 has infinitely many solutions z = ± 1.3169i, where k = 0, ±1, ±2, ... None of these solutions is a real number.

by Tetyana Butler