Trigonometric Functions

1. $\frac{4\pi}{3}$ rad
2. $\frac{−\pi }{3}$ rad
3. $\frac{11\pi}{6}$ rad
4. $210°$
5. $-540°$
6. $-1/2$
7. $-\large\frac{\sqrt{2}}{2}$
8. $\large \frac{\sqrt{3}-1}{2\sqrt{2}} \normalsize = \large \frac{\sqrt{6}-\sqrt{2}}{4}$
   1. $b=5.7$
   2. $\sin A=\frac{4}{7}, \, \cos A=\frac{5.7}{7}, \, \tan A=\frac{4}{5.7}, \, \csc A=\frac{7}{4}, \, \sec A=\frac{7}{5.7}, \, \cot A=\frac{5.7}{4}$
   1. $c=151.7$
   2. $\sin A=0.5623, \, \cos A=0.8273, \, \tan A=0.6797, \, \csc A=1.778, \, \sec A=1.209, \, \cot A=1.471$
   1. $c=85$
   2. $\sin A=\frac{84}{85}, \, \cos A=\frac{13}{85}, \, \tan A=\frac{84}{13}, \, \csc A=\frac{85}{84}, \, \sec A=\frac{85}{13}, \, \cot A=\frac{13}{84}$
   1. $y=\frac{24}{25}$
   2. $\sin \theta =\frac{24}{25}, \, \cos \theta =\frac{7}{25}, \, \tan \theta =\frac{24}{7}, \, \csc \theta =\frac{25}{24}, \, \sec \theta =\frac{25}{7}, \, \cot \theta =\frac{7}{24}$
   1. $x=\frac{−\sqrt{2}}{3}$
   2. $\sin \theta =\frac{\sqrt{7}}{3}, \, \cos \theta =\frac{−\sqrt{2}}{3}, \, \tan \theta =\frac{−\sqrt{14}}{2}, \, \csc \theta =\frac{3\sqrt{7}}{7}, \, \sec \theta =\frac{-3\sqrt{2}}{2}, \, \cot \theta =\frac{−\sqrt{14}}{7}$
9. $\sec^2 x$
10. $\sin^2 x$
11. $\sec^2 \theta$
12. [$\large\frac{1}{\sin t} \normalsize =\csc t$
13.   
14.   
15.   
16.   
17. [$\theta = \{\frac{\pi}{6}, \, \frac{5\pi}{6}\}$
18. $\theta = \{\frac{\pi}{4}, \, \frac{3\pi}{4}, \, \frac{5\pi}{4}, \, \frac{7\pi}{4}\}$
19. $\theta = \{\frac{2\pi}{3}, \, \frac{5\pi}{3}\}$
20. $\theta = \{0, \, \pi, \, \frac{\pi}{3}, \, \frac{5\pi}{3}\}$
21. $y=4\sin\Big(\large\frac{\pi}{4} \normalsize x\Big)$
22. $y=\cos(2\pi x)$
    1. $1$
    2. $2\pi$
    3. $\frac{\pi}{4}$ units to the right
    1. $\frac{1}{2}$
    2. $8\pi$
    3. No phase shift
    1. $3$
    2. $2$
    3. $\frac{2}{\pi}$ units to the left
23. Approximately $42$ in.
    1. $0.550$ rad/sec
    2. $0.236$ rad/sec
    3. $0.698$ rad/min
    4. $1.697$ rad/min
24. $\approx 30.9 \, \text{in}^2$
    1. $\pi /184$; the voltage repeats every $\pi /184$ sec
    2. Approximately $59$ periods
    1. Amplitude = $10$; period = $24$
    2. $47.4^{\circ} F$
    3. $14$ hours later, or 2 p.m.
    4. 

Inverse Functions

1. 1. Not one-to-one

   2. Not one-to-one

   3. One-to-one

   4. 1. $f^{-1}(x)=\sqrt{x+4}$

      2. Domain: $x \ge -4$, Range: $y \ge 0$

   5. 1. $f^{-1}(x)=\sqrt[3]{x-1}$

      2. Domain: all real numbers, Range: all real numbers

   6. 1. $f^{-1}(x)=x^2+1$

      2. Domain: $x \ge 0$, Range: $y \ge 1$

   7. 
   8. 

   9. These are inverses.

   10. These are not inverses.

   11. These are inverses.

   12. These are inverses.

   13. $\frac{\pi}{6}$
   14. $\frac{\pi}{4}$
   15. $\frac{\pi}{6}$
   16. $\frac{\sqrt{2}}{2}$
   17. $-\frac{\pi}{6}$
   18. 1. $x=f^{-1}(V)=\sqrt{0.04-\frac{V}{500}}$

       2. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity $V$.

       3. $0.1$ cm; $0.14$ cm; $0.17$ cm

   19. 1. $$31,250, $66,667, $107,143$
       2. $p=\frac{85C}{C+75}$

       3. $34$ ppb

   20. 1. $~92^{\circ}$
       2. $~42^{\circ}$
       3. $~27^{\circ}$

   21. $x \approx 6.69, 8.51$; so, the temperature occurs on June 21 and August 15

   22. $~1.5 sec$

   23. $\tan^{-1}(\tan(2.1))\approx -1.0416$; the expression does not equal $2.1$ since $2.1>1.57=\frac{\pi}{2}$—in other words, it is not in the restricted domain of $\tan x$. $\cos^{-1}(\cos(2.1))=2.1$, since $2.1$ is in the restricted domain of $\cos x$.

   Exponential and Logarithmic Functions

   1. 1. $125$
      2. $2.24$
      3. $9.74$
   2. 1. $0.01$
      2. $10,000$
      3. $46.42$
   3. b
   4. a
   5. c

   6. Domain: all real numbers, Range: $(2,\infty)$, Horizontal asymptote at $y=2$

   7. Domain: all real numbers, Range: $(0,\infty)$, Horizontal asymptote at $y=0$

   8. Domain: all real numbers, Range: $(-\infty ,1)$, Horizontal asymptote at $y=1$

   9. Domain: all real numbers, Range: $(-1,\infty )$, Horizontal asymptote at $y=-1$

   10. $8^{1/3}=2$
   11. $5^2=25$
   12. $e^{-3}=\frac{1}{e^3}$
   13. $e^0=1$
   14. $\log_4(\frac{1}{16})=-2$
   15. $\log_9 1=0$
   16. $\log_{64} 4=\frac{1}{3}$
   17. $\log_9 150=y$
   18. $\log_4 0.125=-\frac{3}{2}$
   19. 

       Domain: $(1,\infty )$, Range: $(−\infty ,\infty)$, Vertical asymptote at $x=1$

   20. 

       Domain: $(0,\infty)$, Range: $(−\infty ,\infty)$, Vertical asymptote at $x=0$

   21. 

       Domain: $(-1,\infty)$, Range: $(−\infty ,\infty)$, Vertical asymptote at $x=-1$

   22. $2+3\log_3 a-\log_3 b$
   23. $\frac{3}{2}+\frac{1}{2}\log_5 x+\frac{3}{2}\log_5 y$
   24. $-\frac{3}{2}+\ln 6$
   25. $\frac{\ln 15}{3}$
   26. $\frac{3}{2}$
   27. $\log 7.21$
   28. $\frac{2}{3}+\frac{\log 11}{3\log 7}$
   29. $x=\frac{1}{25}$
   30. $x=4$
   31. $x=3$
   32. $1+\sqrt{5}$
   33. $\frac{\ln 82}{\ln 7} \approx 2.2646$
   34. $\frac{\ln 211}{\ln 0.5} \approx -7.7211$
   35. $\frac{\ln 0.452}{\ln 0.2} \approx 0.4934$
   36. $\approx 17,491$

   37. Approximately $$131,653$ is accumulated in $5$ years.

   38. a. $\approx 333$ million b. $94$ years from 2013, or in 2107

   39. 1. a. $k \approx 0.0578$

       2. b. $\approx 92$ hours

   40. The San Francisco earthquake had $10^{3.4}$ or $\approx 2512$ times more energy than the Japan earthquake.