Trigonometry

Trigonmetric 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

θ30°45°60°90°120°180°270°360°
sin θ0$\frac{1}{2}$$\frac{\sqrt{2}}{2}$$\frac{\sqrt{3}}{2}$1$\frac{\sqrt{3}}{2}$0-10
cos θ1$\frac{\sqrt{3}}{2}$$\frac{\sqrt{2}}{2}$$\frac{1}{2}$0$\frac{-1}{2}$-101
tan θ0$\frac{1}{\sqrt{3}}$1$\sqrt{3}$$-\sqrt{3}$00
cot θ$\sqrt{3}$1$\frac{1}{\sqrt{3}}$0$\frac{-1}{\sqrt{3}}$0
sec θ1$\frac{2}{\sqrt{3}}$$\sqrt{2}$2-2-11
cosec θ2$\sqrt{2}$$\frac{2}{\sqrt{3}}$1$\frac{1}{\sqrt{3}}$-1



Identities

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

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

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

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

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

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

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

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

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

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

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

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

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