F.5 student Hoover found it interesting to know some properties of complex trigonometric functions. He asked if the following is still true?
for any complex number .
The answer is affirmative. Here is a verification.
Start with the famous Euler’s formula,
Replace by , yields
By elimination from the equations above, we have
Now, it is a piece of cake to obtain
Could you try to verify another usual properties of real trigonometric functions to see if they are still applicable to the cases in complex numbers? Like say
* may be verified by the power series of the sine function.