Coordinate Definition: In a right-angled triangle with vertex at origin $(0,0)$ and a point $P(x, y)$ on the terminal side with radius $r$ (hypotenuse):

• $\sin \theta = \frac{y}{r}$ (Opposite/Hypotenuse)

• $\cos \theta = \frac{x}{r}$ (Adjacent/Hypotenuse)

• $\tan \theta = \frac{y}{x}$ (Opposite/Adjacent)

Reciprocal Relations:

• $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$

Quotient Relations:

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Pythagorean Identities:

• $\sin^2 \theta + \cos^2 \theta = 1$

• $\sec^2 \theta – \tan^2 \theta = 1$ (implies $\sec^2 \theta = 1 + \tan^2 \theta$)

• $\csc^2 \theta – \cot^2 \theta = 1$ (implies $\csc^2 \theta = 1 + \cot^2 \theta$)