The trigonometric functions can be defined for complex variables as well as real ones.
One way is to use the power series for sin(x) and cos(x), which are convergent for all real and complex numbers. An easier procedure, however, is to use the identities from the previous section:
- cos(i x) = cosh(x)
- sin(i x) = i sinh(x)
- tan(i x) = i tanh(x)
Any complex number z can be written z = x+iy for real x and y. We then have
- sin(z) = sin(x+iy) = sin(x)cos(iy) + cos(x)sin(iy) = sin(x)cosh(y) + icos(x)sinh(y)
- cos(z) = cos(x+iy) = cos(x)cos(iy) - sin(x)sin(iy) = cos(x)cosh(y) - isin(x)sinh(y)
These functions can take any real or complex value, however large. Nevertheless, they still satisfy
- sin2(x) + cos2(x) = 1
Also, it is easily shown that all the results such as sin(z+2π)=sin(z) and sin(-z)=-sin(z) are still true.
Clearly, sin(z) and cos(z) are real if y=0 so z is real. Otherwise, sin(z) is real if cos(x)=0 and cos(z) is real if sin(x)=0.
Since cosh(y) is never zero, and sin(x) and cos(x) are never simultaneously zero, there are no zeroes of sin(z) and cos(z) other than the ones on the real number axis.