Trigonometry 01 (Useful Formulas in Trigonometry)

1. $S=r\theta$  [Arc = Radius $\times$ Angle in radians]
2. Area of a sector $A=\frac{1}{2}r^2\theta$
3. Fundamental identities of trigonometry
• $\cos^2\theta+\sin^2\theta=1$
• $1+\tan^2\theta=\sec^2\theta$
• $\cot^2\theta+1=cosec^2\theta$
4. 
   

   $\theta$	$0^0$	$30^0$	$45^0$	$60^0$	$90^0$	$180^0$
   $\sin \theta$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$0$
   $\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$	$-1$
   $\tan \theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$\infty$	$0$

    
5. $\sin(90^0-\theta)=\cos \theta$
   $\cos(90^0-\theta)=\sin \theta$
    
6. $\sin(-\theta)=-\sin \theta$
   $\cos(-\theta)=\cos \theta$
    
7. $\sin(180^0-\theta)=\sin \theta$
   $\cos(180^0-\theta)=-\cos\theta$
    
8. $\sin(90^0+\theta)=\cos \theta$
   $\cos(90^0+\theta)=-\sin \theta$
    
9. $\sin(180^0+\theta)=-\sin \theta$
   $\cos(180^0+\theta)=-\cos\theta$
    
10. $\sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)$
    $\cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)$
     
11. $\color{green}{\sin(2A)=2\sin(A)\cos(A)}$
     
12. $\color{green}{\cos(2A)=\cos^2(A)-\sin^2(A)=2\cos^2(A)-1=1-2\sin^2(A)}$
     
13. $\color{green}{\tan(2A)=\cfrac{2\tan(A)}{1-\tan^2(A)}}$
     
14. $\color{blue}{\sin(3A)=3\sin(A)-4\sin^3(A)}$
     
15. $\color{blue}{\cos(3A)=4\cos^3(A)-3\cos(A)}$
     
16. $\color{blue}{\tan(3A)=\cfrac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}}$
     
17. $\sin C+\sin D=2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2} \right)$
     
18. $\sin C-\sin D=2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)$
     
19. $\cos C+\cos D=2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$
     
20. $\cos C-\cos D=2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{D-C}{2}\right)$
     
21. $2\sin{ }A \cos{ }B=\sin(A+B)+\sin(A-B)$
22. $2\cos A\sin B=\sin(A+B)-\sin(A-B)$
23. $2\cos A\cos B=\cos(A+B)+\cos(A-B)$
24. $2\sin A\sin B=\cos(A-B)-\cos(A+B)$
     
25. If $\sin\theta=\sin\alpha$, then $\theta=n\pi+(-1)^n\alpha$
    If $\cos\theta=\cos\alpha$, then $\theta=2n\pi\pm\alpha$
    If $\tan\theta=\tan\alpha$, then $\theta=n\pi+\alpha$
    Here $n\in\mathbb{Z}$
     
26. For an $ABC$ triangle, in the standard notation,
1. $\cfrac{\sin A}{a}=\cfrac{\sin B}{b}=\cfrac{\sin C}{c}$
2. $\cos A=\cfrac{b^2+c^2-a^2}{2bc}$
3. $a=b$ $\cos C+c$ $\cos B$
4. $\tan\left(\cfrac{B-C}{2}\right)=\left(\cfrac{b-c}{b+c}\right)\cot\cfrac{A}{2}$
5. $\cos\cfrac{A}{2}=\sqrt{\cfrac{s(s-a)}{bc}}$
   Here $s$ is the double of perimeter, of the $ABC$ triangle $(2s=a+b+c)$