## Exercises - Solving Trigonometric Equations

1. Find all solutions of the following equations:

1. $\tan x = 0$

2. $2 \cos x + \sqrt{2} = 0$

3. $\cos^2 x - 1 = 0$

4. $2\cos^2 x - 3\cos x - 2 = 0$

5. $\tan^2 x + (\sqrt{3} - 1)\tan x - \sqrt{3} = 0$

6. $3\sec^2 x = \sec x$

7. $2\sin^2 x - \sin x - 1 = 0$

8. $\cos 2x = \sin x$

9. $\displaystyle{\frac{1+\cos x}{\cos x} = 2}$

10. $\displaystyle{\sqrt{\frac{1+2 \sin x}{2}} = 1}$

11. $\displaystyle{\cos^3 x - \cos x = 0}$

12. $\displaystyle{2 \cos 3x = 1}$

13. $\displaystyle{\cos^2 \theta + \sin \theta = \frac{5}{4}}$

14. $\displaystyle{2\sec^2 x - 5\tan x - 3 = 0}$

See full solutions.

1. $\pi n, \quad n \in \mathbb{Z}$

2. $\pm \frac{\pi}{4} + 2\pi n, \quad n \in \mathbb{Z}$

3. $\pi n, \quad n \in \mathbb{Z}$

4. $\pm \frac{2\pi}{3} + 2\pi n, \quad n \in \mathbb{Z}$

5. $\displaystyle{\left. \begin{array}{c} \frac{\pi}{4} + \pi n\\ \frac{2\pi}{3} + \pi n \end{array} \right\} \quad n \in \mathbb{Z}}$

6. no solutions

7. $\frac{\pi}{2} + \frac{2\pi}{3} n, \quad n \in \mathbb{Z}$

8. $\frac{\pi}{6} + \frac{2\pi}{3} n, \quad n \in \mathbb{Z}$

9. $2\pi n, \quad n \in \mathbb{Z}$

10. $\displaystyle{\left. \begin{array}{c} \frac{\pi}{6} + 2\pi n\\ \frac{5\pi}{6} + 2\pi n \end{array} \right\} \quad n \in \mathbb{Z}}$

11. $\frac{\pi}{2}n, \quad n \in \mathbb{Z}$

12. $\pm \frac{\pi}{9} + \frac{2\pi}{3} n, \quad n \in \mathbb{Z}$

13. $\displaystyle{\left. \begin{array}{c} \frac{\pi}{6} + 2\pi n\\ \frac{5\pi}{6} + 2\pi n \end{array} \right\} \quad n \in \mathbb{Z}}$

14. $\displaystyle{\left. \begin{array}{c} \textrm{arctan}\left(\frac{5+\sqrt{33}}{4}\right)+\pi n\\ \textrm{arctan}\left(\frac{5-\sqrt{33}}{4}\right)+\pi n \end{array} \right\} \quad n \in \mathbb{Z}}$