Trigonometric Ratios of an Acute Angle
In a right-angled triangle $\Delta ABC$ right-angled at $B$:
$$\sin \theta = \frac{\text{Opposite Side (Perpendicular)}}{\text{Hypotenuse}} = \frac{p}{h}$$
$$\cos \theta = \frac{\text{Adjacent Side (Base)}}{\text{Hypotenuse}} = \frac{b}{h}$$
$$\tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{\sin \theta}{\cos \theta}$$
$$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}$$
Specific Angle Values Table
| Angle $\theta$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin \theta$ | $0$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | $1$ |
| $\cos \theta$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | $0$ |
| $\tan \theta$ | $0$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | $\text{Not Defined}$ |
Core Trigonometric Identities
- $\sin^2\theta + \cos^2\theta = 1$
- $1 + \tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1$
- $1 + \cot^2\theta = \csc^2\theta \implies \csc^2\theta - \cot^2\theta = 1$
