Sine, Cosine, and Tangent

In a right triangle with angle $\theta$, the three core trigonometric ratios are

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}.$

They extend to all real angles via the unit circle: $\sin\theta$ is the y-coordinate of the point on the unit circle at angle $\theta$ from the positive x-axis, $\cos\theta$ is the x-coordinate, and $\tan\theta$ is their ratio.

Sin and cos are bounded between $-1$ and $1$; both are periodic with period $2\pi$. Tan has vertical asymptotes wherever $\cos\theta = 0$ (i.e. at $\theta = \pi/2 + k\pi$).

These three functions describe wave behaviour (sound, light, ocean swells), rotational motion, alternating current, and Fourier decomposition — they are arguably the most reused functions in all of physics and engineering.