Trigonometric Identities

Trigonometric Identities

1. Relations of functions

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

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

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

   $\csc \varphi =\frac{1}{\sin \varphi }$

   
    

2. Pythagorean Formula

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

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

   
    

3. Trigonometric functions in terms of their complements

   $\sin \varphi =\cos (\frac{\pi }{2}-\varphi )$

   $\cos \varphi =\sin (\frac{\pi }{2}-\varphi )$

   $\tan \varphi =\cot (\frac{\pi }{2}-\varphi )$

   $\cot \varphi =\tan (\frac{\pi }{2}-\varphi )$

   $\sec \varphi =\csc (\frac{\pi }{2}-\varphi )$

   $\csc \varphi =\sec (\frac{\pi }{2}-\varphi )$

   
    

4. Trigonometric functions in terms of their supplements

   $\sin (\pi -\theta )=\sin \theta$

   $\cos (\pi -\theta )=-\cos \theta$

   $\tan (\pi -\theta )=-\tan \theta$

   
    

5. Trigonometric Periodicity Identities

   $\sin \varphi =\sin (\pi + 2\pi )$

   $\cos \varphi =\cos (\pi + 2\pi )$

   $\tan \varphi =\tan (\pi + \pi )$

   $\cot \varphi =\cot (\pi + \pi )$

   $\sec \varphi =\sec (\pi + 2\pi )$

   $\csc \varphi =\csc (\pi + 2\pi )$

   consequently...

   $\sin \varphi =-\sin (\pi + \pi )$

   $\cos \varphi =-\cos (\pi + \pi )$

   $\csc \varphi =-\csc (\pi + \pi )$

   $\sec \varphi =-\sec (\pi + \pi )$

   
    

6. Negative Angle Indentities

   $\sin (-\varphi ) =-\sin \varphi$

   $\cos (-\varphi ) =\cos \varphi$

   $\tan (-\varphi ) =-\tan \varphi$

   
    

7. The sum and difference of two angles

   $\sin (\alpha +\beta ) =\sin \alpha \cos \beta + \cos \alpha \sin \beta$

   $\cos (\alpha +\beta ) =\cos \alpha \cos \beta - \sin \alpha \sin \beta$

   $\sin (\alpha -\beta ) =\sin \alpha \cos \beta - \cos \alpha \sin \beta$

   $\cos (\alpha -\beta ) =\cos \alpha \cos \beta + \sin \alpha \sin \beta$

   $\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$

   $\tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }$

   
    

8. Double angle formulas

   $\sin 2\varphi = 2\sin \varphi \cos \varphi$

   $\cos 2\varphi = {\cos }^2\varphi - {\sin }^2\varphi$

   $\cos 2\varphi = 2{\cos }^2\varphi - 1$

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

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

   
    

9. Half angle formulas

   $\sin \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{2}}$

   $\cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}$

   $\tan \frac{\theta }{2}=\frac{\sin \theta }{1+\cos \theta }=\frac{1-\cos \theta }{\sin \theta }$