# 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 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

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

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

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

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

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

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

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

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

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

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

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

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

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