In one line
For complex \(z\), \(\sin z=\dfrac{e^{iz}-e^{-iz}}{2i}\) and \(\cos z=\dfrac{e^{iz}+e^{-iz}}{2}\); these come from Euler's formula and, unlike the real case, are not limited to the range \([-1,1]\).
Explained simply
For a real angle, \(\sin\) and \(\cos\) are lengths in a right triangle, so they stay between \(-1\) and 1. A complex number is not an angle you can draw in a triangle, so we need a new definition that agrees with the old one for real angles and still works for complex inputs. Euler's formula gives exactly that. It is like a new phone app that opens your old files and also new formats.
From \(e^{i\theta}=\cos\theta+i\sin\theta\) and \(e^{-i\theta}=\cos\theta-i\sin\theta\), adding and subtracting gives \(\cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}2\), \(\sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}\). We now define \(\sin z,\cos z\) by these same formulas for every complex \(z\). The other circular functions follow: \(\tan z=\dfrac{\sin z}{\cos z}\), \(\sec z=\dfrac1{\cos z}\), and so on.
How it works
- Start from Euler's formula for real \(\theta\).
- Solve for \(\cos\theta\) and \(\sin\theta\) in terms of exponentials.
- Replace \(\theta\) by complex \(z\) to define \(\sin z\), \(\cos z\).
- Check that all usual identities still hold, e.g. \(\sin^2z+\cos^2z=1\), \(\sin(z_1+z_2)=\sin z_1\cos z_2+\cos z_1\sin z_2\). They do, because the proofs use only exponent rules.
Key formula
\(e^{iz}=\cos z+i\sin z\), \(\ \cos z=\dfrac{e^{iz}+e^{-iz}}2\), \(\ \sin z=\dfrac{e^{iz}-e^{-iz}}{2i}\), where \(z=x+iy\) is any complex number and \(e\approx2.718\). There are no units.
Quick example
Find \(\cos i\). Put \(z=i\): \(iz=i^2=-1\). So \(\cos i=\dfrac{e^{-1}+e^{1}}2=\dfrac{0.3679+2.7183}{2}\approx1.543\).
This is bigger than 1, which is impossible for a real angle but perfectly fine for a complex one. Similarly \(\sin i=\dfrac{e^{-1}-e}{2i}=i\,\dfrac{e-e^{-1}}{2}\approx1.175i\).
Common mistakes
- Assuming \(|\sin z|\le1\) for complex \(z\); that bound is only for real arguments.
- Forgetting the \(2i\) in the denominator of \(\sin z\), or writing \(2\) instead.
- Mixing up \(e^{iz}\) with \(e^{i}\cdot e^{z}\); the exponent is the product \(iz\).
How MU asks it
Usually a short 3-mark part or the first line of a longer proof: "Using the exponential definitions, prove \(\sin^2z+\cos^2z=1\)" or "Show that \(\cos i\) is real and greater than 1". For full marks write Euler's formula, the two exponential definitions, substitute, and simplify cleanly. These definitions are the base for every separation-of-parts question in this module.