trigonometry

Formulas

1. Square Sum Formulas
   • ${\sin }^{2}x+{\cos }^{2}x=1$
   • ${\tan }^{2}x={\sec }^{2}x-1$
   • ${\cot }^{2}x={\csc }^{2}x-1$
2. Compound Angle Formulas
   • $\sin (A+B)=\sin (A)\cos (B)+\cos (A)\sin (B)$
   • $\sin (A-B)=\sin (A)\cos (B)-\cos (A)\sin (B)$
   • $\cos (A+B)=\cos (A)\cos (B)-\sin (A)\sin (B)$
   • $\cos (A-B)=\cos (A)\cos (B)+\sin (A)\sin (B)$
   • $\tan (A+B)=\frac{\tan (A)+\tan (B)}{1-\tan (A)\tan (B)}$
   • $\tan (A-B)=\frac{\tan (A)-\tan (B)}{1+\tan (A)\tan (B)}$
   • $\cot (A+B)=\frac{\cot (A)\cot (B)-1}{\cot (A)+\cot (B)}$
   • $\cot (A-B)=\frac{\cot (A)\cot (B)+1}{\cot (A)-\cot (B)}$
   • $\sin (A+B)\times \sin (A-B)={\sin }^2(A)-{\sin }^2(B)$
   • $\cos (A+B)\times \cos (A-B)={\cos }^2(A)-{\sin }^2(B)={\cos }^2(B)-{\sin }^2(A)$
3. Transformation Formulas
   • $\sin (x)+\sin (y)=2\sin (\frac{x+y}{2})\cos (\frac{x-y}{2})$
   • $\sin (x)-\sin (y)=2\cos (\frac{x+y}{2})\sin (\frac{x-y}{2})$
   • $\cos (x)+\cos (y)=2\cos (\frac{x+y}{2})\cos (\frac{x-y}{2})$
   • $\cos (x)-\cos (y)=-2\sin (\frac{x+y}{2})\sin (\frac{x-y}{2})=2\sin (\frac{x+y}{2})\sin (\frac{y-x}{2})$
   • $2\sin (A)cos(B)=\sin (A+B)+\sin (A-B)$
   • $2\cos (A)sin(B)=\sin (A+B)-\sin (A-B)$
   • $2\cos (A)cos(B)=\cos (A+B)+\cos (A-B)$
   • $2\sin (A)sin(B)=\cos (A-B)-\cos (A+B)$
4. Series
   • $\sin (a)+\sin (a+d)+\sin (a+2d)+.....+\sin (a+(n-1)d)=\frac{\sin (\frac{nd}{2})}{\sin (\frac{d}{2})}\times \sin (a+(n-1)\frac{d}{2})$
   • $\cos (a)+\cos (a+d)+\cos (a+2d)+.....+\cos (a+(n-1)d)=\frac{\sin (\frac{nd}{2})}{\sin (\frac{d}{2})}\times \cos (a+(n-1)\frac{d}{2})$
   • $\cos (x)\times \cos (2x)\times \cos (2^{2}x)\times .....\times \cos (2^{n-1}x)=\frac{\sin (2^{n}x)}{2^n\sin (x)}$
5. If A + B + C = $\pi$
   • $\sin (2A)+\sin (2B)+\sin (2C)=4\sin (A)\sin (B)\sin (C)$
   • $\cos (2A)+\cos (2B)+\cos (2C)=-1-4\cos (A)\cos (B)\cos (C)$
   • $\sin (A)+\sin (B)+\sin (C)=4\cos (\frac{A}{2})\cos (\frac{B}{2})\cos (\frac{C}{2})$
   • $\tan (A)+\tan (B)+\tan (C)=\tan (A)\tan (B)\tan (C)$
   • $\tan (\frac{A}{2})\tan (\frac{B}{2})+\tan (\frac{B}{2})\tan (\frac{C}{2})+\tan (\frac{C}{2})\tan (\frac{A}{2})=1$
6. If A+B=C $\tan (A)\tan (B)\tan (C)=\tan (A)-\tan (B)-\tan (C)$
7. Multiple Angle Formulas
   • $\sin (2x)=2\sin (x)\cos (x)=\frac{2\tan (x)}{1+{\tan }^2(x)}$
   • $\cos (2x)={\cos }^2(x)-{\sin }^2(x)=\frac{1-ta{n}^2(x)}{1+{\tan }^2(x)}$
   • $\tan (2x)=\frac{2\tan (x)}{1-{\tan }^2(x)}$
   • $\cos (2x)=2{\cos }^2(x)-1=1-2{\sin }^2(x)$
   • $\sin (3x)=3\sin (x)-4{\sin }^3(x)$
   • $\cos (3x)=4{\sin }^3(x)-3\sin (x)$
   • $\tan (3x)=\frac{3\tan (x)-{\tan }^3(x)}{1-3{\tan }^2(x)}$
8. Multiple Angle Formulas (Extra)
   • $(\cos (x)+\sin (x){)}^2=1+\sin (2x)$
   • $\frac{1-\cos (2x)}{\sin (2x)}=\tan (x)$
   • $\frac{1-\cos (2x)}{1+\cos (2x)}={\tan }^2(x)$
   • $\frac{\tan (x)+\tan (y)}{\tan (x)-\tan (y)}=\frac{\sin (x+y)}{\sin (x-y)}$
   • $\tan (\frac{\pi }{4}+x)=\frac{1+\tan (x)}{1-\tan (x)}=\frac{\cos (x)+\sin (x)}{\cos (x)-\sin (x)}$
   • $\frac{\cos (2x)}{1+\sin (2x)}=\tan (\frac{\pi }{4}-x)$
   • $\cot (x)-\tan (x)=2\cot (2x)$
9. Trigonometric Equations Observations
   • $\sin (x)=0=>x=n\pi$
   • $\cos (x)=0=>x=(2n+1)\frac{\pi }{2}$
   • $\cos (x)=1=>x=2n\pi$
   • $\cos (x)=-1=>x=(2n+1)\pi$
10. Trigonometric Equations Results
    • $\sin (x)=\sin (y)=>x=n\pi +(-1{)}^{n}y$
    • $\cos (x)=\cos (y)=>x=2n\pi \pm y$
    • $\tan (x)=\tan (y)=>x=n\pi \pm y$
    • For the below 3 result below $x=n\pi \pm y$
      • ${\sin }^2(x)={\sin }^2(y)$
      • ${\cos }^2(x)={\cos }^2(y)$
      • ${\tan }^2(x)={\tan }^2(y)$
11. General Solutions

    Trigonometric Equations	General Solutions
    $\sin \theta =0$	$\theta =n\pi$
    $\cos \theta =0$	$\theta =(n\pi +\pi /2)$
    $\tan \theta =0$	$\theta =n\pi$
    $\sin \theta =1$	$\theta =(2n\pi +\pi /2)=(4n+1)\pi /2$
    $\cos \theta =1$	$\theta =2n\pi$
    $\sin \theta =\sin \alpha$	$\theta =n\pi +(-1{)}^n\alpha \text{ where }\alpha \in [-\pi /2,\pi /2]$
    $\cos \theta =\cos \alpha$	$\theta =2n\pi \pm \alpha \text{ where }\alpha \in [0,\pi ]$
    $\tan \theta =\tan \alpha$	$\theta =n\pi +\alpha \text{ where }\alpha \in [-\pi /2,\pi /2]$
    $\sin 2\theta =\sin 2\alpha$	$\theta =n\pi \pm \alpha$
    $\cos 2\theta =\cos 2\alpha$	$\theta =n\pi \pm \alpha$
    $\tan 2\theta =\tan 2\alpha$	$\theta =n\pi \pm \alpha$