Geometry Complex Numbers and Trigonometry

Being tired of memorizing these formulas? See how a unit circle and the complex numbers can help.

Trigonometry - you think it is about triangles, but really it is about circles.

Consider a unit circle (whose radius is 1), the coordinator of a point (z) on the circle can be written as $(\cos\theta,\sin\theta)$, where $\theta$ is the angle the line of oz rotates from x axis clockwise, so to rotate another angle $\phi$, we will have $(\cos(\theta+\phi),\sin(\theta+\phi)$.


Now, instead of y axis, think of it as an axis with unit of $i$, where $i=\sqrt{-1}$.

Then the coordinator can be written as $(\cos(\theta+\phi),i\sin(\theta+\phi)$, which is actually the result two rotation.

Thus we have:


then it is not hard to derive that:




For example, $\cos75^o=\cos(45^o+30^o)$:

Consider it geometrically, is the x axis rotates from $0^o$ first $45^o$ then another $30^o$.

Thus we can easily write


where the real part

$\cos45^o\cos30^o-\sin45^o\sin30^o$ is $cos75^o$,

and the imaginary part (without i)

$cos45^o\sin30^o+\sin45^o\cos30^o$ is $\sin75^o$.


- YouTube: Complex number fundamentals | Lockdown math ep. 3