Exercises - Complex Numbers Birth Trigonometry!

1. Draw the given angle as a rotation about the origin in the Cartesian plane:

1. $60^{\circ}$

2. $135^{\circ}$

3. $1140^{\circ}$

4. $-240^{\circ}$

5. $\frac{\pi}{3}$

6. $\frac{7\pi}{6}$

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

8. $\frac{5\pi}{2}$

2. Convert from degrees to radians:

1. $45^{\circ}$

2. $270^{\circ}$

3. $1^{\circ}$

4. $-230^{\circ}$

5. $540^{\circ}$

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

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

3. $\frac{\pi}{180}$

4. $-\frac{23\pi}{18}$

5. $3\pi$

3. Convert from radians to degrees:

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

2. $7\pi$

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

4. $\frac{19\pi}{2}$

1. $120^{\circ}$

2. $1260^{\circ}$

3. $30^{\circ}$

4. $1710^{\circ}$

4. Find the values of the following:

1. $\cos 5\pi$

2. $\sin(-\frac{7\pi}{6})$

3. $\cos \frac{23\pi}{4}$

4. $\sin 9\pi$

5. $\cos (-\frac{10\pi}{3})$

6. $\sin (-\frac{4\pi}{3})$

7. $\cot \frac{13\pi}{6}$

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

9. $\csc (-\frac{\pi}{6})$

10. $\tan \frac{23\pi}{4}$

11. $\sec \frac{10\pi}{3}$

12. $\csc 5\pi$

13. $\cot (-\frac{5\pi}{4})$

14. $\sin 150^{\circ}$

15. $\sec (-120^{\circ})$

16. $\csc 495^{\circ}$

1. $-1$

2. $\frac{1}{2}$

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

4. $0$

5. $-\frac{1}{2}$

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

7. $\sqrt{3}$

8. does not exist (not in the domain)

9. $-2$

10. $-1$

11. $-2$

12. does not exist (not in the domain)

13. $-1$

14. $\frac{1}{2}$

15. $-2$

16. $\sqrt{2}$

5. Make a table giving the values of all six trigonometric functions for $t = 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4},\frac{5\pi}{4},\pi,\frac{7\pi}{6},\frac{5\pi}{4},\frac{4\pi}{3},\frac{3\pi}{2},\frac{5\pi}{3},\frac{7\pi}{4},\frac{11\pi}{6},\textrm{ and } 2\pi$.

6. Find the value of

1. $\sin \theta$, if $\cot \theta = \frac{3}{4}$ and $\pi \lt \theta \lt \frac{3\pi}{2}$

2. $\sec \theta$, if $\csc \theta = -3$ and $\frac{3\pi}{2} \lt \theta \lt 2\pi$

3. $\cos t$, if $\tan t = -\frac{2}{3}$ and $\frac{3\pi}{2} \lt t \lt 2\pi$

4. $\sin t$, if $\sec t = \frac{13}{5}$ and $0 \lt t \lt \frac{\pi}{2}$

5. $\tan t$, if $\csc t = \frac{5}{3}$ and $\frac{\pi}{2} \lt t \lt \pi$

6. $\cot t$, if $\csc t = \frac{5}{4}$ and $0 \lt t \lt \frac{\pi}{2}$

7. $\sin \theta$, if $\cot \theta = -\frac{4}{9}$ and $\frac{\pi}{2} \lt \theta \lt \pi$

8. $\cos \theta$, if $\tan \theta = \frac{\sqrt{3}}{2}$ and $\pi \lt \theta \lt \frac{3\pi}{2}$

9. $\sec \theta$, if $\sin \theta = -\frac{1}{6}$ and $\frac{3\pi}{2} \lt \theta \lt 2\pi$

10. $\csc \theta$, if $\cot \theta = -\frac{\sqrt{13}}{12}$ and $\frac{\pi}{2} \lt \theta \lt \pi$

1. As $\cot \theta = \frac{3}{4}$, we know $\frac{\cos \theta}{\sin \theta} = \frac{3}{4}$, so if $x = \sin \theta$, then $\cos \theta = \frac{3x}{4}$. By the Pythagorean identity, $x^2 + \frac{9x^2}{16} = 1$. Solving for $x$ yields $\pm \frac{4}{5}$, but the restriction that $\pi \lt \theta \lt \frac{3\pi}{2}$ tells us $\theta$ is in quadrant III, where the sine is negative. Thus, $\sin \theta = -\frac{4}{5}$

2. As $\csc \theta = -3$, we know $\sin \theta = -\frac{1}{3}$. Then by the Pythagorean identity, $\cos^2 \theta + \frac{1}{9} = 1$. Solving for $\cos \theta$, we find $\cos \theta = \pm \sqrt{1 - \frac{1}{9}} = \pm \sqrt{\frac{8}{9}} = \pm \frac{2\sqrt{2}}{3}$. Thus, $\sec \theta = \pm \frac{3}{2\sqrt{2}} = \pm \frac{3\sqrt{2}}{4}$. Noting the restriction that $\frac{3\pi}{2} \lt \theta \lt 2\pi$, we realize $\theta$ is in quadrant IV, where the cosine (and hence, secant) are positive. Therefore, $\sec \theta = \frac{3\sqrt{2}}{4}$.

3. $\frac{3\sqrt{13}}{13}$

4. $\frac{12}{13}$

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

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

7. $\frac{9\sqrt{97}}{97}$

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

9. $\frac{6\sqrt{35}}{35}$

10. $\frac{\sqrt{157}}{12}$

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

1.   $\displaystyle{\frac{\sin \theta}{\cos \theta} = 1 - \frac{\cos \theta}{\sin \theta}}$

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

3.   $\displaystyle{1 - 2\sin^2 \theta = 2\cos^2 \theta - 1}$

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

5.   $\displaystyle{\frac{\sec^4 t - \tan^4 t}{1 - 2\tan^2 t} = 1}$

6.   $\displaystyle{\sin^2 \theta \cot^2 \theta + \cos^2 \theta \tan^2 \theta = 1}$

7.   $\displaystyle{\sec \theta - \frac{\cos \theta}{1 + \sin \theta} = \cot \theta}$

8.   $\displaystyle{\frac{\tan^2 x}{1 + \cos x} = \frac{\sec x - 1}{\cos x}}$

9.   $\displaystyle{(\csc t - \cot t)^2 = \frac{1 - \cos t}{1 + \cos t}}$

10.   $\displaystyle{1 + \frac{1}{\cos \theta} = \frac{\tan^2 \theta}{\sec \theta - 1}}$

1. not an identity

2. identity

3. identity

4. not an identity

5. not an identity

6. identity

7. not an identity

8. identity

9. identity

10. identity

8. Graph the following for $-2\pi \le x \le 2\pi$.

1. $y = 4\cos x$

2. $y = \sin \frac{2}{3} x$

3. $y = 4\cos(2x - \frac{3\pi}{2})$

4. $y = \sin(x - \frac{\pi}{6})$

5. $y= -\frac{1}{2} \sin x$

9. Graph the following.

1. $y = -\frac{8}{5} \cos (\frac{x}{5} + \frac{\pi}{3})$ over $[-5\pi,10\pi]$

2. $y = 4\sin(2x - \frac{\pi}{6})$ over $[-\pi,2\pi]$

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

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

10. Graph the following.

1. $y = 1 + \cos x$ for $-2\pi \le x \le 2\pi$

2. $y = 2 - \sin x$ from $-\pi$ to $\frac{3\pi}{2}$

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

4. $y = 2 - 3\cos 2x$ over $[-2\pi,\pi]$   (omit finding the $x$-intercepts)

11. Graph the following. Label interecepts and other important features (e.g., asymptotes)

1. $y = -\tan x$ over $[-2\pi,2\pi]$

2. $y = -\sec x$ from $-\pi$ to $\pi$

3. $y = \frac{1}{2} \tan 2x$ from $-\frac{5\pi}{4}$ to $\frac{3\pi}{8}$

4. $y = \csc 3x$ for $-\frac{\pi}{2} \le x \le \frac{5\pi}{6}$

5. $y = 2\tan \frac{x}{2}$ from $-3\pi$ to $\frac{5\pi}{2}$

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

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

1. $0 \pm \pi n, \quad (n = 0, 1, 2, \ldots)$

2. $\displaystyle{\left. \begin{array}{c} \frac{3\pi}{4} \pm 2\pi n\\ \frac{5\pi}{4} \pm 2\pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$

3. $\displaystyle{\left. \begin{array}{c} 0 \pm 2\pi n\\ \pi \pm 2\pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$   or more compactly,   $\pm \pi n, \quad (n = 0, 1, 2, \ldots)$

4. $\displaystyle{\left. \begin{array}{c} \frac{2\pi}{3} \pm 2\pi n\\ \frac{4\pi}{3} \pm 2\pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$

5. $\displaystyle{\left. \begin{array}{c} \frac{\pi}{4} \pm \pi n\\ \frac{2\pi}{3} \pm \pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$

6. no solutions

7. $\displaystyle{\left. \begin{array}{c} \frac{7\pi}{6} \pm 2\pi n\\ \frac{11\pi}{6} \pm 2\pi n\\ \frac{\pi}{2} \pm 2\pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$

8. $\displaystyle{\left. \begin{array}{c} \frac{\pi}{6} \pm 2\pi n\\ \frac{5\pi}{6} \pm 2\pi n\\ \frac{3\pi}{2} \pm 2\pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$

9. $0 \pm 2\pi n, \quad (n = 0, 1, 2, \ldots)$

10. $\displaystyle{\left. \begin{array}{c} \frac{\pi}{6} \pm \pi n\\ \frac{5\pi}{6} \pm \pi n \end{array} \right\} \quad (n = 0, 1, 2, \ldots)}$

13. Find the values of the following:

1. $\arcsin 0$

2. $\textrm{arccot}\, (-\frac{\sqrt{3}}{3})$

3. $\cot(\textrm{arccot}\,(-3))$

4. $\cos(\arccos \frac{4}{5})$

5. $\textrm{arccsc}\, 2$

6. $\arccos (-1)$

7. $\csc(\arcsin \frac{3}{5})$

8. $\textrm{arcsec}\, (\sin \frac{\pi}{2})$

9. $\sin (\textrm{arcsec}\, 2)$

10. $\arccos (-\frac{1}{2})$

11. $\arctan^3 (-\sqrt{3})$

12. $3\arcsin^2 (\frac{\sqrt{3}}{2})$

13. $\textrm{arcsec}\, 0$

14. $\sin(\arctan 2)$

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

16. $\tan(\arccos(-\frac{2}{3}))$

17. $\arccos 2$

18. $\cos(\arcsin(-\frac{4}{5}))$

19. $4 \arctan 1$

20. $\csc(\textrm{arcsec}\, 12)$

21. $\textrm{arccsc}\, \sqrt{2}$

22. $\textrm{arcsec}\, 2$

23. $\arctan(\sin \frac{\pi}{2})$

24. $\arctan(\cos \pi)$

25. $\arcsin (\tan \frac{\pi}{4})$

1. $0$

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

3. $-3$

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

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

6. $\pi$

7. $\frac{5}{3}$

8. $0$

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

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

11. $-\frac{\pi^3}{27}$

12. $\frac{\pi^2}{3}$

13. no value

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

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

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

17. no value

18. $\frac{3}{5}$

19. $\pi$

20. $\frac{12}{\sqrt{143}}$

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

22. $\frac{\pi}{3}$

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

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

25. $\frac{\pi}{2}$

14. Write the given expression in terms of $x$ without any trigonometric functions:

1. $\sin(\arctan x)$

2. $\tan(\arcsin x)$

3. $\cot(\arcsin x)$

4. $\cos(\arcsin x)$

5. $\cos(\textrm{arcsec}\, x)$

6. $\csc(\textrm{arccot}\, \frac{x}{4})$

1. $\displaystyle{\frac{x}{\sqrt{x^2 + 1}}}$

2. $\displaystyle{\frac{x}{\sqrt{1-x^2}}}$

3. $\displaystyle{\frac{\sqrt{1-x^2}}{x}}$

4. $\displaystyle{\sqrt{1-x^2}}$

5. $\displaystyle{\frac{1}{x}}$

6. $\displaystyle{\frac{\sqrt{x^2 + 16}}{4}}$