## Exercises - Review of Trigonometry

1. Find the value of each of the following: $\newcommand {\arccos}{\textrm{arccos}\,}$ $\newcommand {\arcsin}{\textrm{arcsin}\,}$ $\newcommand {\arctan}{\textrm{arctan}\,}$ $\newcommand {\arcsec}{\textrm{arcsec}\,}$ $\newcommand {\arccsc}{\textrm{arccsc}\,}$ $\newcommand {\arccot}{\textrm{arccot}\,}$

1. $\tan \left( -\frac{7\pi}{6} \right)$

2. $\cot \theta, \textrm{ if } \sin \theta = \frac{2}{3} \textrm{ and } \frac{\pi}{2} \lt \theta \lt \pi$

3. $\cos \left( \frac{-11\pi}{6} \right)$

4. $\csc t, \textrm{ if } \cos t = -\frac{15}{17} \textrm{ and } \pi \lt t \lt \frac{3\pi}{2}$

5. $\tan \left(-\frac{\pi}{6} \right)$

6. $\arccos \left( \sin \frac{11\pi}{4} \right)$

7. $\cos(\arctan(-1))$

8. $\sec(\arccot(\frac{-5}{12}))$

9. $\arcsec(-\sqrt{2})$

10. $\sec(\arccot \frac{x}{3})$

11. $\arctan(\sin(-\frac{5\pi}{2}))$

12. $\sec(-\frac{5\pi}{6})$

13. $9\arccot^2 \frac{\sqrt{3}}{3}$

14. $\arccsc(-2)$

15. $\tan(\arcsin(-\frac{3}{5}))$

16. $\arccos(\cot \frac{11\pi}{4})$

17. $\csc t, \textrm{ if } \tan t = 3 \textrm{ and } \pi \lt t \lt \frac{3\pi}{2}$

18. $\arccsc(\cos(\frac{-\pi}{3}))$

19. $\tan \theta, \textrm{ if } \sin \theta = \frac{5}{13} \textrm{ and } \frac{\pi}{2} \lt \theta \lt \pi$

20. $\tan \frac{9\pi}{2}$

21. $\cos(\arctan \frac{x}{3})$

22. $\arcsin(\cos \frac{\pi}{2})$

23. $\arcsec 1$

24. $\csc(-\frac{9\pi}{4})$

25. $\cos(\arcsin x)$

26. $\tan \frac{54\pi}{6}$

See full solutions.

1. $-\frac{\sqrt{3}}{3}$

2. $-\frac{\sqrt{5}}{2}$

3. $\frac{\sqrt{3}}{2}$

4. $-\frac{17}{8}$

5. $-\frac{\sqrt{3}}{3}$

6. $\frac{\pi}{4}$

7. $\frac{\sqrt{2}}{2}$

8. $-\frac{13}{5}$

9. $\frac{3\pi}{4}$

10. $\frac{\sqrt{x^2+9}}{x}$

11. $-\frac{\pi}{4}$

12. $-\frac{2}{\sqrt{3}}$

13. $\pi^2$

14. $-\frac{\pi}{6}$

15. $-\frac{3}{4}$

16. $\pi$

17. $-\frac{\sqrt{10}}{3}$

18. no value

19. $-\frac{5}{12}$

20. no value

21. $\frac{3}{\sqrt{x^2 + 9}}$

22. $0$

23. $0$

24. $-\sqrt{2}$

25. $\sqrt{1-x^2}$

26. $0$

2. Find the value of each of the following: $\newcommand {\arccos}{\textrm{arccos}\,}$ $\newcommand {\arcsin}{\textrm{arcsin}\,}$ $\newcommand {\arctan}{\textrm{arctan}\,}$ $\newcommand {\arcsec}{\textrm{arcsec}\,}$ $\newcommand {\arccsc}{\textrm{arccsc}\,}$ $\newcommand {\arccot}{\textrm{arccot}\,}$

1. $\arcsec^2 (\csc \frac{2\pi}{3})$

2. $\cos^2 \frac{7\pi}{6}$

3. $\arccsc(-\sqrt{2})$

4. $5\arccot(-\frac{\sqrt{3}}{3})$

5. $\cot(\arcsin(-\frac{8}{17}))$

6. $\arcsin(\sin \frac{5\pi}{3})$

7. $\cos(\arctan 5)$

8. $\arccos(\sin(-\frac{\pi}{6}))$

9. $\sin t, \textrm{ if } \cot t = -\frac{12}{5} \textrm{ and } \frac{3\pi}{2} \lt t \lt 2\pi$

10. $\cos(\arctan(\frac{-3}{4}))$

11. $\arcsec(-\frac{2\sqrt{3}}{3})$

12. $\arccos(\sin \frac{23\pi}{4})$

13. $\cot t, \textrm{ if } \sec t = -\frac{8}{5} \textrm{ and } \pi \lt t \lt \frac{3\pi}{2}$

See full solutions.

1. $\frac{\pi^2}{36}$

2. $\frac{3}{4}$

3. $-\frac{\pi}{4}$

4. $\frac{2\pi}{3}$

5. $-\frac{15}{8}$

6. $-\frac{\pi}{3}$

7. $\frac{\sqrt{26}}{26}$

8. $\frac{2\pi}{3}$

9. $-\frac{5}{13}$

10. $\frac{4}{5}$

11. $\frac{5\pi}{6}$

12. $\frac{3\pi}{4}$

13. $\frac{5\sqrt{39}}{39}$

3. Sketch graphs of the following. Label intercepts, asymptotes, and endpoints.

1. $y = -4\cos(\frac{x}{2} - \frac{\pi}{4})$ on $[-\pi,3\pi]$

2. $y = 4\sin(\frac{2}{3} x + \frac{\pi}{6})$ from $-2\pi$ to $2\pi$

3. $y = -2\sec(3x+\pi)$ on $[-\frac{\pi}{3},\frac{2\pi}{3}]$

4. $y = 2\csc(2x + \frac{\pi}{2})$ for $-\frac{\pi}{2} \le x \le \pi$

5. $y = 5\cos(3x + \frac{\pi}{2})$ on $[0,\pi]$

6. $y = 4 - 2\sin(\frac{x}{3}-\frac{\pi}{3})$ from $-2\pi$ to $\frac{5\pi}{2}$

7. $y = \tan(x + \frac{\pi}{4})$ on $[-\pi,2\pi]$

8. $y = -2\sin(x - \frac{3\pi}{2})$ on $[-\frac{\pi}{2},2\pi]$

9. $y = \cot(x - \frac{\pi}{6})$ from $-\pi$ to $2\pi$

10. $y = -\frac{1}{3} \cos(\frac{x}{2} + \frac{\pi}{4})$ on $[-2\pi,4\pi]$

11. $y = -\sec \frac{x}{2}$ for $-3\pi \lt x \lt 3\pi$

4. Show whether each of the following is or is not an identity:

1. $\displaystyle{\frac{1}{\csc \theta + \cot \theta} + \frac{\sec \theta + 1}{\tan \theta} = 2\csc \theta}$

2. $\displaystyle{\frac{1+\tan \theta}{1-\tan \theta} + \frac{1 + \cot \theta}{1 - \cot \theta} = 0}$

3. $\displaystyle{\frac{1}{1+\cos \theta} - \frac{1}{1-\cos \theta} = \frac{2}{\sec \theta - \cos \theta}}$

4. $\displaystyle{(1+\sec x)(1-\cos x) = \tan x \sin x}$

5. $\displaystyle{\sec 2\theta = \frac{\sec^2 \theta}{2 - \sec^2 \theta}}$

6. $\displaystyle{\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta}$

7. $\displaystyle{\frac{\tan^2 x \csc^2 x - 1}{\csc x \tan^2 x \sin x} = 1}$

8. $\displaystyle{\frac{1}{\sec \theta - \tan \theta} = \sec \theta + \tan \theta}$

9. $\displaystyle{\frac{\sin \theta + \tan \theta}{1 + \cos \theta} = \cot \theta}$

10. $\displaystyle{\cos^4 x - \sin^4 x = \cos 2x}$

11. $\displaystyle{\frac{\tan t}{\tan^2 t - 1} = \frac{1}{\tan t - \cot t}}$

12. $\displaystyle{\sec x - \sin x \tan x = \cos x}$

13. $\displaystyle{\frac{\sin \theta + \cos \theta}{\sec \theta + \csc \theta} = \frac{\sin \theta}{\sec \theta}}$

14. $\displaystyle{\frac{\cos^4 x - \sin^4 x}{\sin x + \cos x} = \frac{1-\tan x}{1+\tan x}}$

Note, (c), (i), (n) are not identities; the others are identities

See full solutions.

5. Find all solutions of the following equations:

1. $\displaystyle{\tan^2 x + \sec^2 x + 3\sec x = 1}$

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

3. $\displaystyle{2\sin x \tan x + \tan x - 2\sin x - 1 = 0}$

4. $\displaystyle{2\cos^2 x - 3\cos x + 1 = 0}$

5. $\displaystyle{3\tan x + \frac{1}{\tan x} = 2\sqrt{3}}$

6. $\displaystyle{2\sin x \tan x = 3}$

7. $\displaystyle{\sec^2 x - \tan x = 1}$

8. $\displaystyle{4 \sin^2 x - 1 = 0}$

9. $\displaystyle{\cos 2x = \cos x}$

10. $\displaystyle{2\cos^3 x + \sin^2 x = 1}$

11. $\displaystyle{8\sin^4 x - 10 \sin^2 x + 3 = 0}$

12. $\displaystyle{2\sin^2 x + 7\sin x + 3 = 0}$

13. $\displaystyle{3 \cot x = \tan x}$

See the full solutions.

Note: the condition that $n=0,1,2,3,\ldots$ should be considered attached to all answers below:

1. $\frac{2\pi}{3} \pm 2\pi n; \frac{4\pi}{3} \pm 2\pi n$

2. $0 \pm \pi n; \frac{7\pi}{6} \pm \pi n; \frac{11\pi}{6} \pm \pi n$

3. $\frac{7\pi}{6} \pm 2\pi n; \frac{11\pi}{6} \pm 2\pi n; \frac{\pi}{4} \pm \pi n$

4. $\frac{\pi}{3} \pm 2\pi n; \frac{5\pi}{3} \pm 2\pi n; 0 \pm 2\pi n$

5. $\frac{\pi}{6} \pm \pi n$

6. $\frac{\pi}{3} \pm 2\pi n; \frac{5\pi}{3} \pm 2\pi n$

7. $0 \pm \pi n; \frac{\pi}{4} \pm \pi n$

8. $\frac{\pi}{6} \pm \pi n; \frac{5\pi}{6} \pm \pi n$

9. $\frac{2\pi}{3} \pm 2\pi n; \frac{4\pi}{3} \pm 2\pi n; 0 \pm 2\pi n$

10. $\frac{\pi}{2} \pm \pi n; \frac{\pi}{3} \pm 2\pi n; \frac{5\pi}{3} \pm 2\pi n$

11. $\frac{\pi}{4} \pm \pi n; \frac{3\pi}{4} \pm \pi n; \frac{\pi}{3} \pm \pi n; \frac{2\pi}{3} \pm \pi n$

12. $\frac{7\pi}{6} \pm 2\pi n; \frac{11\pi}{6} \pm 2\pi n$

13. $\frac{\pi}{3} \pm \pi n; \frac{2\pi}{3} \pm \pi n$

6. Verify the following identities:

1. $\displaystyle{\frac{1}{\tan x + \cot x} = (\sin x)(\cos x)}$

2. $\displaystyle{\sec^2 \alpha + \csc^2 \alpha = \tan^2 \alpha + \cot^2 \alpha + 2}$

3. $\displaystyle{\frac{\tan x - \sin x}{\tan x + \sin x} = \frac{\sec x -1}{\sec x + 1}}$

4. $\displaystyle{(1 - \cos^2 \theta)(1+ \cot^2 \theta) = 1}$

5. $\displaystyle{\frac{csc \beta - \sin \beta}{1 - \sin^2 \beta} = \csc \beta}$

6. $\displaystyle{\frac{\sin \gamma \sec^2 \gamma - \sin \gamma}{\cos \gamma} = \tan^3 \gamma}$

7. $\displaystyle{\frac{\cot^2 \theta - 1}{1 - \tan^2 \theta} = \cot^2 \theta}$

8. $\displaystyle{\frac{\tan \alpha}{\sec \alpha + 1} = \frac{1}{\cot \alpha + \csc \alpha}}$

9. $\displaystyle{\frac{\cos^2 x + 3\cos x + 2}{\sin^2 x} = \frac{2 + \cos x}{1 - \cos x}}$

10. $\displaystyle{\frac{1+\sin x + \cos x}{1+\cos x - \sin x} = \sec x + \tan x}$     (challenge!)