Complex trigonometric functions

Euler's formula \(e^{iy}=\cos y+i\sin y\) and its conjugate \(e^{-iy}=\cos y-i\sin y\) can be solved for sine and cosine: adding and subtracting gives exactly the expressions in the box. They are taken as the definition for any \(z\), and agree with the real functions when \(z\) is real. The other functions are defined as usual: \(\tan z=\sin z/\cos z\), and so on.

Almost every real identity survives: \(\sin^2z+\cos^2z=1\), \(\sin(z+w)=\sin z\cos w+\cos z\sin w\), \(\sin(z+2\pi)=\sin z\) and \(\sin(z+\pi)=-\sin z\). They are checked by substituting the exponentials, so knowing the properties of \(e^{z}\) is enough. The zeros are also the familiar ones: \(\sin z=0\) only at \(z=k\pi\), and \(\cos z=0\) only at \(z=\frac{\pi}{2}+k\pi\).

What changes is the size. Writing \(z=x+iy\) gives \(\sin z=\sin x\cosh y+i\cos x\sinh y\) and \(\cos z=\cos x\cosh y-i\sin x\sinh y\). Since \(\cosh y\) and \(\sinh y\) grow without bound, neither sine nor cosine is bounded in the plane, and for instance \(\cos(iy)=\cosh y\) takes the value 2. The inequality \(|\sin z|\le1\) is a property of the real line only.

A common mistake is to treat the hyperbolic functions as a separate topic: in fact \(\cos(iy)=\cosh y\) and \(\sin(iy)=i\sinh y\), so trigonometric and hyperbolic functions are one family seen along different axes. This will be used when studying entire functions and when solving equations such as \(\cos z=w\).