# Maths Resources - Trignometry Identities

Trignometry Identities

### Trignometric Ratios for Right Triangle

$$\sin {\theta} = \frac {p}{h} = \frac {perpendicular}{hypotenuse} $$

$$\cos {\theta} = \frac {b}{h} = \frac {base}{hypotenuse} $$

$$\tan {\theta} = \frac {p}{b} = \frac {perpendicular}{base} $$

$$\cot {\theta} = \frac {b}{p} = \frac {base}{perpendicular} $$

$$\sec {\theta} = \frac {h}{b} = \frac {hypotenuse}{base} $$

$$\csc {\theta} = \frac {h}{p} = \frac {hypotenuse}{perpendicular} $$

### Trignometric Table

θ | 0° | 30° | 45° | 60° | 90° | 120° | 180° | 270° | 360° |
---|---|---|---|---|---|---|---|---|---|

sin θ | 0 | $$\frac{1}{2}$$ | $$\frac{\sqrt{2}}{2}$$ | $$\frac{\sqrt{3}}{2}$$ | 1 | $$\frac{\sqrt{3}}{2}$$ | 0 | -1 | 0 |

cos θ | 1 | $$\frac{\sqrt{3}}{2}$$ | $$\frac{\sqrt{2}}{2}$$ | $$\frac{1}{2}$$ | 0 | $$\frac{-1}{2}$$ | -1 | 0 | 1 |

tan θ | 0 | $$\frac{1}{\sqrt{3}}$$ | 1 | $$\sqrt{3}$$ | ∞ | $$-\sqrt{3}$$ | 0 | ∞ | 0 |

cot θ | ∞ | $$\sqrt{3}$$ | 1 | $$\frac{1}{\sqrt{3}}$$ | 0 | $$\frac{-1}{\sqrt{3}}$$ | ∞ | 0 | ∞ |

sec θ | 1 | $$\frac{2}{\sqrt{3}}$$ | $$\sqrt{2}$$ | 2 | ∞ | -2 | -1 | ∞ | 1 |

cosec θ | ∞ | 2 | $$\sqrt{2}$$ | $$\frac{2}{\sqrt{3}}$$ | 1 | $$\frac{1}{\sqrt{3}}$$ | ∞ | -1 | ∞ |

### Identities

$$\tan {A} = \frac{\sin{A}}{\cos{A}} $$

$$\sec {A} = \frac{1}{\cos{A}}$$

$$cosec A = \frac{1}{\sin{A}}$$

$$\cot {A} = \frac{1}{\tan{A}} = = \frac{\cos {A}}{\sin{A}}$$

$$\sin^2{A} + \cos^2{A} = 1 $$

$$\sec^2{A} = 1 + \tan^2{A}$$

$$\csc^2{A} = 1 + \cot^2{A}$$

$$\sin{A\pm B} = \sin{A}\cos{B}\pm \cos{A}\sin{B}$$

$$\cos{A\pm B} = \cos{A}\cos{B}\mp \sin{A}\sin{B}$$

$$\tan{A\pm B} = \frac{\tan{A}\pm \tan{B}}{\tan{A}\mp \tan{B}}$$

$$\sin{2A} = 2\sin{A}\cos{A} $$

$$\cos{2A} = \cos^2{A} - \sin^2{A} = 2\cos^2{A} - 1 = 1 - 2\sin^2{A} $$

$$\tan{2A} = \frac{2\tan{A}}{1 - \tan^2{A}} $$