Trigonometric Identities: Period, Symmetry & Sign
Understanding how trigonometric functions behave under shifts and reflections
1. Periodicity Identities
When we rotate a radius by angle $\alpha$ on the unit circle, we arrive at a point $B(x_0, y_0)$. If we rotate by $\alpha + 2\pi n$ (where $n$ is any integer), we complete $n$ full revolutions and land on exactly the same point $B$.
Since the coordinates are the same, the trig function values are the same:
$$\sin(\alpha + 2\pi n) = \sin \alpha$$
$$\cos(\alpha + 2\pi n) = \cos \alpha$$
$$\mathrm{tg}(\alpha + 2\pi n) = \mathrm{tg}\,\alpha$$
$$\mathrm{ctg}(\alpha + 2\pi n) = \mathrm{ctg}\,\alpha$$
Key fact: The period of sine and cosine is $2\pi$ (or $360°$). This is the smallest positive number $T$ such that $f(\alpha + T) = f(\alpha)$ for all $\alpha$.