Plane Trigonometry

The Six Trigonometric Functions
Right triangle with sides a, b, and c and angle theta1. $\sin \theta = \dfrac{a}{c}$

2. $\cos \theta = \dfrac{b}{c}$

3. $\tan \theta = \dfrac{a}{b}$

4. $\csc \theta = \dfrac{c}{a}$

5. $\sec \theta = \dfrac{c}{b}$

6. $\cot \theta = \dfrac{b}{a}$
 

Trigonometric Identities
1. $\sin \theta = \dfrac{1}{\csc \theta}$

2. $\cos \theta = \dfrac{1}{\sec \theta}$

3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta}$

4. $\cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$

5. $\sec \theta = \dfrac{1}{\cos \theta}$

6. $\csc \theta = \dfrac{1}{\sin \theta}$

7. $\sin^2 \theta + \cos^2 \theta = 1$

8. $\tan^2 \theta + 1 = \sec^2 \theta$

9. $1 + \cot^2 \theta = \csc^2 \theta$

10. $\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B$

11. $\sin (A - B) = \sin A \, \cos B - \cos A \, \sin B$

12. $\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B$

13. $\cos (A - B) = \cos A \, \cos B + \sin A \, \sin B$

14. $\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B}$

15. $\tan (A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \, \tan B}$

16. $\sin 2\theta = 2 \sin \theta \, \cos \theta$

17. $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2\sin^2 \theta = 2\cos^2 \theta - 1$

18. $\tan 2\theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$

19. $\sin \frac{1}{2}\theta = \sqrt{\dfrac{1 - \cos \theta}{2}}$

20. $\cos \frac{1}{2}\theta = \sqrt{\dfrac{1 + \cos \theta}{2}}$

21. $\tan \frac{1}{2}\theta = \dfrac{1 - \cos \theta}{\sin \theta} = \dfrac{\sin \theta}{1 + \cos \theta} = \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}}$

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