Trignometric Formulas

Trignometry

Compound Angle Formulas

• $\sin (A+B)=\sin A\cdot \cos B+\cos A\cdot \sin B$
• $\sin (A-B)=\sin A\cdot \cos B-\cos A\cdot \sin B$
• $\cos (A+B)=\cos A\cdot \cos B-\sin A\cdot \sin B$
• $\cos (A-B)=\cos A\cdot \cos B+\sin A\cdot \sin B$
• $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
• $\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\cdot \tan B}$
• $\cot (A+B)=\frac{\cot B\cdot \cot A-1}{\cot B+\cot A}$
• $\cot (A-B)=\frac{\cot B\cdot \cot A+1}{\cot B-\cot A}$

Sum to Product Formulas (C & D)

• $\sin C+\sin D=2\cdot \sin \left(\frac{C+D}{2}\right)\cdot \cos \left(\frac{C-D}{2}\right)$
• $\sin C-\sin D=2\cos \left(\frac{C+D}{2}\right)\cdot \sin \left(\frac{C-D}{2}\right)$
• $\cos C+\cos D=2\cos \left(\frac{C+D}{2}\right)\cdot \cos \left(\frac{C-D}{2}\right)$
• $\cos C-\cos D=-2\sin \left(\frac{C+D}{2}\right)\cdot \sin \left(\frac{C-D}{2}\right)=2\sin \left(\frac{C+D}{2}\right)\cdot \sin \left(\frac{D-C}{2}\right)$

Multiple Angle Formulas

Double Angle

• $\sin 2x=2\sin x\cdot \cos x=\frac{2\tan x}{1+{\tan }^{2}x}$
• $\cos 2x={\cos }^{2}x-{\sin }^{2}x=2{\cos }^{2}x-1=1-2{\sin }^{2}x=\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}$
• $\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}$

Triple Angle

• $\sin 3x=3\sin x-4{\sin }^{3}x$
• $\cos 3x=4{\cos }^{3}x-3\cos x$
• $\tan 3x=\frac{3\tan x-{\tan }^{3}x}{1-3{\tan }^{2}x}$

Maximum and Minimum Values

• $a\sin x+b\cos x$
  • Max Value $=+\sqrt{a^2+b^2}$
  • Min Value $=-\sqrt{a^2+b^2}$

Even and Odd Functions & Negative Angles

• Odd $f(x)\Rightarrow f(-x)=-f(x)$
• Even $f(x)\Rightarrow f(-x)=+f(x)$
• $\cos (-x)=\cos x$
• $\sin (-x)=-\sin x$
• $\tan (-x)=-\tan x$
• $\sec (-x)=\sec x$
• $\mathrm{cosec}(-x)=-\mathrm{cosec}x$
• $\cot (-x)=-\cot x$

Product to Sum Formulas

• $2\sin A\cos B=\sin (A+B)+\sin (A-B)$
• $2\cos A\sin B=\sin (A+B)-\sin (A-B)$
• $2\cos A\cos B=\cos (A+B)+\cos (A-B)$
• $-2\sin A\sin B=\cos (A+B)-\cos (A-B)$

Difference of Squares Formulas

• $\sin (A+B)\cdot \sin (A-B)={\sin }^{2}A-{\sin }^{2}B$
• $\cos (A+B)\cdot \cos (A-B)={\cos }^{2}A-{\sin }^{2}B$

Fundamental Identities & Variations

• ${\sin }^2\theta +{\cos }^2\theta =1$
• ${\sec }^2\theta -{\tan }^2\theta =1$
• ${\mathrm{cosec}}^2\theta -{\cot }^2\theta =1$
• $1+\cos 2x=2{\cos }^{2}x$
• $1-\cos 2x=2{\sin }^{2}x$
• $\tan x=\frac{1-\cos 2x}{\sin 2x}$

General Solution of trigonometry

• $\sin x=0\Rightarrow x=n\pi$
• $\cos x=0\Rightarrow x=(2n+1)\pi /2$
• $\tan x=0\Rightarrow x=n\pi$
• $\sin x=\sin y\Rightarrow x=n\pi +(-1{)}^n\cdot y$
• $\cos x=\cos y\Rightarrow x=2n\pi \pm y$
• $\tan x=\tan y\Rightarrow x=n\pi +y$

Squared Equations

• ${\sin }^{2}x={\sin }^{2}y$
• ${\cos }^{2}x={\cos }^{2}y$
• ${\tan }^{2}x={\tan }^{2}y$
  All imply $\Rightarrow x=n\pi \pm y$
  $\boxed{n\in \mathbb{Z}}$