3.7 Section Exercises

Verbal

1. We know $\text{ }g\left(x\right)=\mathrm{cos}\text{ }x\text{ }$ is an even function, and $\text{ }f\left(x\right)=\mathrm{sin}\text{ }x\text{ }$ and $\text{ }h\left(x\right)=\mathrm{tan}\text{ }x\text{ }$ are odd functions. What about $\text{ }G\left(x\right)={\mathrm{cos}}^{2}x,F\left(x\right)={\mathrm{sin}}^{2}x,$ and $\text{ }H\left(x\right)={\mathrm{tan}}^{2}x?\text{ }$ Are they even, odd, or neither? Why?

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All three functions, $\text{ }F,G,$ and $\text{ }H,$ are even.

This is because $\text{ }F\left(-x\right)=\mathrm{sin}\left(-x\right)\mathrm{sin}\left(-x\right)=\left(-\mathrm{sin}\text{ }x\right)\left(-\mathrm{sin}\text{ }x\right)={\mathrm{sin}}^{2}x=F\left(x\right),G\left(-x\right)=\mathrm{cos}\left(-x\right)\mathrm{cos}\left(-x\right)=\mathrm{cos}\text{ }x\mathrm{cos}\text{ }x={\mathrm{cos}}^{2}x=G\left(x\right)\text{ }$ and $\text{ }H\left(-x\right)=\mathrm{tan}\left(-x\right)\mathrm{tan}\left(-x\right)=\left(-\mathrm{tan}\text{ }x\right)\left(-\mathrm{tan}\text{ }x\right)={\mathrm{tan}}^{2}x=H\left(x\right).$

2. Examine the graph of $\text{ }f\left(x\right)=\mathrm{sec}\text{ }x\text{ }$ on the interval $\text{ }\left[-\pi ,\pi \right].\text{ }$ How can we tell whether the function is even or odd by only observing the graph of $\text{ }f\left(x\right)=\mathrm{sec}\text{ }x?$

3. After examining the reciprocal identity for $\text{ }\mathrm{sec}\text{ }t,$ explain why the function is undefined at certain points.

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When $\text{ }\mathrm{cos}\text{ }t=0,$ then $\text{ }\mathrm{sec}\text{ }t=\frac{1}{0},$ which is undefined.

4. All of the Pythagorean identities are related. Describe how to manipulate the equations to get from $\text{ }{\mathrm{sin}}^{2}t+{\mathrm{cos}}^{2}t=1\text{ }$ to the other forms.

Algebraic

For the following exercises, use the fundamental identities to fully simplify the expression.

5. $\mathrm{sin}\text{ }x\text{ }\mathrm{cos}\text{ }x\text{ }\mathrm{sec}\text{ }x$

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$\mathrm{sin}\text{ }x$

6. $\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-x\right)\mathrm{csc}\left(-x\right)$

7. $\mathrm{tan}\text{ }x\mathrm{sin}\text{ }x+\mathrm{sec}\text{ }x{\mathrm{cos}}^{2}x$

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$\mathrm{sec}\text{ }x$

8. $\mathrm{csc}\text{ }x+\mathrm{cos}\text{ }x\mathrm{cot}\left(-x\right)$

9. $\frac{\mathrm{cot}\text{ }t+\mathrm{tan}\text{ }t}{\mathrm{sec}\left(-t\right)}$

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$\mathrm{csc}\text{ }t$

10. $3\text{ }{\mathrm{sin}}^{3}\text{ }t\text{ }\mathrm{csc}\text{ }t+{\mathrm{cos}}^{2}\text{ }t+2\text{ }\mathrm{cos}\left(-t\right)\mathrm{cos}\text{ }t$

11. $-\mathrm{tan}\left(-x\right)\mathrm{cot}\left(-x\right)$

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$-1$

12. $\frac{-\mathrm{sin}\left(-x\right)\mathrm{cos}\text{ }x\text{ }\mathrm{sec}\text{ }x\text{ }\mathrm{csc}\text{ }x\text{ }\mathrm{tan}\text{ }x}{\mathrm{cot}\text{ }x}$

13. $\frac{1+{\mathrm{tan}}^{2}\theta }{{\mathrm{csc}}^{2}\theta }+{\mathrm{sin}}^{2}\theta +\frac{1}{{\mathrm{sec}}^{2}\theta }$

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${\mathrm{sec}}^{2}x$

14. $\left(\frac{\mathrm{tan}\text{ }x}{{\mathrm{csc}}^{2}x}+\frac{\mathrm{tan}\text{ }x}{{\mathrm{sec}}^{2}x}\right)\left(\frac{1+\mathrm{tan}\text{ }x}{1+\mathrm{cot}\text{ }x}\right)-\frac{1}{{\mathrm{cos}}^{2}x}$

15. $\frac{1-{\mathrm{cos}}^{2}\text{ }x}{{\mathrm{tan}}^{2}\text{ }x}+2\text{ }{\mathrm{sin}}^{2}\text{ }x$

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${\mathrm{sin}}^{2}x+1$

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

16. $\frac{\mathrm{tan}\text{ }x+\mathrm{cot}\text{ }x}{\mathrm{csc}\text{ }x};\text{ }\mathrm{cos}\text{ }x$

17. $\frac{\mathrm{sec}\text{ }x+\mathrm{csc}\text{ }x}{1+\mathrm{tan}\text{ }x};\text{ }\mathrm{sin}\text{ }x$

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$\frac{1}{\mathrm{sin}\text{ }x}$

18. $\frac{\mathrm{cos}\text{ }x}{1+\mathrm{sin}\text{ }x}+\mathrm{tan}\text{ }x;\text{ }\mathrm{cos}\text{ }x$

19. $\frac{1}{\mathrm{sin}\text{ }x\mathrm{cos}\text{ }x}-\mathrm{cot}\text{ }x;\text{ }\mathrm{cot}\text{ }x$

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$\frac{1}{\mathrm{cot}\text{ }x}$

20. $\frac{1}{1-\mathrm{cos}\text{ }x}-\frac{\mathrm{cos}\text{ }x}{1+\mathrm{cos}\text{ }x};\text{ }\mathrm{csc}\text{ }x$

21. $\left(\mathrm{sec}\text{ }x+\mathrm{csc}\text{ }x\right)\left(\mathrm{sin}\text{ }x+\mathrm{cos}\text{ }x\right)-2-\mathrm{cot}\text{ }x;\text{ }\mathrm{tan}\text{ }x$

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$\mathrm{tan}\text{ }x$

22. $\frac{1}{\mathrm{csc}\text{ }x-\mathrm{sin}\text{ }x};\text{ }\mathrm{sec}\text{ }x\text{ and }\mathrm{tan}\text{ }x$

23. $\frac{1-\mathrm{sin}\text{ }x}{1+\mathrm{sin}\text{ }x}-\frac{1+\mathrm{sin}\text{ }x}{1-\mathrm{sin}\text{ }x};\text{ }\mathrm{sec}\text{ }x\text{ and }\mathrm{tan}\text{ }x$

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$-4\mathrm{sec}\text{ }x\mathrm{tan}\text{ }x$

24. $\mathrm{tan}\text{ }x;\text{ }\mathrm{sec}\text{ }x$

25. $\mathrm{sec}\text{ }x;\text{ }\mathrm{cot}\text{ }x$

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$±\sqrt{\frac{1}{{\mathrm{cot}}^{2}x}+1}$

26. $\mathrm{sec}\text{ }x;\text{ }\mathrm{sin}\text{ }x$

27. $\mathrm{cot}\text{ }x;\text{ }\mathrm{sin}\text{ }x$

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$\frac{±\sqrt{1-{\mathrm{sin}}^{2}x}}{\mathrm{sin}\text{ }x}$

28. $\mathrm{cot}\text{ }x;\text{ }\mathrm{csc}\text{ }x$

For the following exercises, verify the identity.

29. $\mathrm{cos}\text{ }x-{\mathrm{cos}}^{3}x=\mathrm{cos}\text{ }x\text{ }{\mathrm{sin}}^{2}\text{ }x$

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Answers will vary. Sample proof:

$\mathrm{cos}\text{ }x-{\mathrm{cos}}^{3}x=\mathrm{cos}\text{ }x\left(1-{\mathrm{cos}}^{2}x\right)$ $=\mathrm{cos}\phantom{\rule{0.2em}{0ex}}x{\mathrm{sin}}^{2}x$

30. $\mathrm{cos}\text{ }x\left(\mathrm{tan}\text{ }x-\mathrm{sec}\left(-x\right)\right)=\mathrm{sin}\text{ }x-1$

31. $\frac{1+{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}=\frac{1}{{\mathrm{cos}}^{2}x}+\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}=1+2\text{ }{\mathrm{tan}}^{2}x$

Show Solution

Answers will vary. Sample proof:

$\frac{1+{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}=\frac{1}{{\mathrm{cos}}^{2}x}+\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}={\mathrm{sec}}^{2}x+{\mathrm{tan}}^{2}x={\mathrm{tan}}^{2}x+1+{\mathrm{tan}}^{2}x=1+2{\mathrm{tan}}^{2}x$

32. ${\left(\mathrm{sin}\text{ }x+\mathrm{cos}\text{ }x\right)}^{2}=1+2\text{ }\mathrm{sin}\text{ }x\mathrm{cos}\text{ }x$

33. ${\mathrm{cos}}^{2}x-{\mathrm{tan}}^{2}x=2-{\mathrm{sin}}^{2}x-{\mathrm{sec}}^{2}x$

Show Solution

Answers will vary. Sample proof:

${\mathrm{cos}}^{2}x-{\mathrm{tan}}^{2}x=1-{\mathrm{sin}}^{2}x-\left({\mathrm{sec}}^{2}x-1\right)=1-{\mathrm{sin}}^{2}x-{\mathrm{sec}}^{2}x+1=2-{\mathrm{sin}}^{2}x-{\mathrm{sec}}^{2}x$

Extensions

For the following exercises, prove or disprove the identity.

34. $\frac{1}{1+\mathrm{cos}\text{ }x}-\frac{1}{1-\mathrm{cos}\left(-x\right)}=-2\text{ }\mathrm{cot}\text{ }x\text{ }\mathrm{csc}\text{ }x$

35. ${\mathrm{csc}}^{2}x\left(1+{\mathrm{sin}}^{2}x\right)={\mathrm{cot}}^{2}x$

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36. $\left(\frac{{\mathrm{sec}}^{2}\left(-x\right)-{\mathrm{tan}}^{2}x}{\mathrm{tan}\text{ }x}\right)\left(\frac{2+2\text{ }\mathrm{tan}\text{ }x}{2+2\text{ }\mathrm{cot}\text{ }x}\right)-2\text{ }{\mathrm{sin}}^{2}x=\mathrm{cos}\text{ }2x$

37. $\frac{\mathrm{tan}\text{ }x}{\mathrm{sec}\text{ }x}\mathrm{sin}\left(-x\right)={\mathrm{cos}}^{2}x$

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38. $\frac{\mathrm{sec}\left(-x\right)}{\mathrm{tan}\text{ }x+\mathrm{cot}\text{ }x}=-\mathrm{sin}\left(-x\right)$

39. $\frac{1+\mathrm{sin}\text{ }x}{\mathrm{cos}\text{ }x}=\frac{\mathrm{cos}\text{ }x}{1+\mathrm{sin}\left(-x\right)}$

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For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

40. $\frac{{\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta }{1-{\mathrm{tan}}^{2}\theta }={\mathrm{sin}}^{2}\theta$

41. $3\text{ }{\mathrm{sin}}^{2}\theta +4\text{ }{\mathrm{cos}}^{2}\theta =3+{\mathrm{cos}}^{2}\theta$

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True $\text{ }3\text{ }{\mathrm{sin}}^{2}\theta +4\text{ }{\mathrm{cos}}^{2}\theta =3\text{ }{\mathrm{sin}}^{2}\theta +3\text{ }{\mathrm{cos}}^{2}\theta +{\mathrm{cos}}^{2}\theta =3\left({\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta \right)+{\mathrm{cos}}^{2}\theta =3+{\mathrm{cos}}^{2}\theta$

42. $\frac{\mathrm{sec}\text{ }\theta +\mathrm{tan}\text{ }\theta }{\mathrm{cot}\text{ }\theta +\mathrm{cos}\text{ }\theta }={\mathrm{sec}}^{2}\theta$

3.7 More Section Exercises

Verbal

1. Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.

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There will not always be solutions to trigonometric function equations. For a basic example, $\text{ }\mathrm{cos}\left(x\right)=-5.$

2. When solving a trigonometric equation involving more than one trig function, do we always want to try to rewrite the equation so it is expressed in terms of one trigonometric function? Why or why not?

3. When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions?

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Algebraic

For the following exercises, find all solutions exactly on the interval $\text{ }0\le \theta <2\pi .$

4. $2\text{ }\mathrm{sin}\text{ }\theta =-\sqrt{2}$

5. $2\text{ }\mathrm{sin}\text{ }\theta =\sqrt{3}$

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

6. $2\text{ }\mathrm{cos}\text{ }\theta =1$

7. $2\text{ }\mathrm{cos}\text{ }\theta =-\sqrt{2}$

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$\frac{3\pi }{4},\frac{5\pi }{4}$

8. $\mathrm{tan}\text{ }\theta =-1$

9. $\mathrm{tan}\text{ }x=1$

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$\frac{\pi }{4},\frac{5\pi }{4}$

10. $\mathrm{cot}\text{ }x+1=0$

11. $4\text{ }{\mathrm{sin}}^{2}x-2=0$

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$\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

12. ${\mathrm{csc}}^{2}x-4=0$

For the following exercises, solve exactly on $\text{ }\left[0,2\pi \right).$

13. $2\text{ }\mathrm{cos}\text{ }\theta =\sqrt{2}$

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$\frac{\pi }{4},\frac{7\pi }{4}$

14. $2\text{ }\mathrm{cos}\text{ }\theta =-1$

15. $2\text{ }\mathrm{sin}\text{ }\theta =-1$

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$\frac{7\pi }{6},\frac{11\pi }{6}$

16. $2\text{ }\mathrm{sin}\text{ }\theta =-\sqrt{3}$

17. $2\text{ }\mathrm{sin}\left(3\theta \right)=1$

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$\frac{\pi }{18},\frac{5\pi }{18},\frac{13\pi }{18},\frac{17\pi }{18},\frac{25\pi }{18},\frac{29\pi }{18}$

18. $2\text{ }\mathrm{sin}\left(2\theta \right)=\sqrt{3}$

19. $2\text{ }\mathrm{cos}\left(3\theta \right)=-\sqrt{2}$

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$\frac{3\pi }{12},\frac{5\pi }{12},\frac{11\pi }{12},\frac{13\pi }{12},\frac{19\pi }{12},\frac{21\pi }{12}$

20. $\mathrm{cos}\left(2\theta \right)=-\frac{\sqrt{3}}{2}$

21. $2\text{ }\mathrm{sin}\left(\pi \theta \right)=1$

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$\frac{1}{6},\frac{5}{6},\frac{13}{6},\frac{17}{6},\frac{25}{6},\frac{29}{6},\frac{37}{6}$

22. $2\text{ }\mathrm{cos}\left(\frac{\pi }{5}\theta \right)=\sqrt{3}$

For the following exercises, find all exact solutions on $\text{ }\left[0,2\pi \right).$

23. $\mathrm{sec}\left(x\right)\mathrm{sin}\left(x\right)-2\text{ }\mathrm{sin}\left(x\right)=0$

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$0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}$

24. $\mathrm{tan}\left(x\right)-2\text{ }\mathrm{sin}\left(x\right)\mathrm{tan}\left(x\right)=0$

25. $2\text{ }{\mathrm{cos}}^{2}t+\mathrm{cos}\left(t\right)=1$

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

26. $2\text{ }{\mathrm{tan}}^{2}\left(t\right)=3\text{ }\mathrm{sec}\left(t\right)$

27. $2\text{ }\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)-\mathrm{sin}\left(x\right)+2\text{ }\mathrm{cos}\left(x\right)-1=0$

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$\frac{\pi }{3},\frac{3\pi }{2},\frac{5\pi }{3}$

28. ${\mathrm{cos}}^{2}\theta =\frac{1}{2}$

29. ${\mathrm{sec}}^{2}x=1$

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$0,\pi$

30. ${\mathrm{tan}}^{2}\left(x\right)=-1+2\text{ }\mathrm{tan}\left(-x\right)$

31. $8\text{ }{\mathrm{sin}}^{2}\left(x\right)+6\text{ }\mathrm{sin}\left(x\right)+1=0$

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$\pi -{\mathrm{sin}}^{-1}\left(-\frac{1}{4}\right),\frac{7\pi }{6},\frac{11\pi }{6},2\pi +{\mathrm{sin}}^{-1}\left(-\frac{1}{4}\right)$

32. ${\mathrm{tan}}^{5}\left(x\right)=\mathrm{tan}\left(x\right)$

For the following exercises, solve with the methods shown in this section exactly on the interval $\text{ }\left[0,2\pi \right).$

33. $\mathrm{sin}\left(3x\right)\mathrm{cos}\left(6x\right)-\mathrm{cos}\left(3x\right)\mathrm{sin}\left(6x\right)=-0.9$

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$\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{\pi }{3}-\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{2\pi }{3}+\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\pi -\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{4\pi }{3}+\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{5\pi }{3}-\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right)$

34. $\mathrm{sin}\left(6x\right)\mathrm{cos}\left(11x\right)-\mathrm{cos}\left(6x\right)\mathrm{sin}\left(11x\right)=-0.1$

35. $\mathrm{cos}\left(2x\right)\mathrm{cos}\text{ }x+\mathrm{sin}\left(2x\right)\mathrm{sin}\text{ }x=1$

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$0$

36. $6\text{ }\mathrm{sin}\left(2t\right)+9\text{ }\mathrm{sin}\text{ }t=0$

37. $9\text{ }\mathrm{cos}\left(2\theta \right)=9\text{ }{\mathrm{cos}}^{2}\theta -4$

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$\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6}$

38. $\mathrm{sin}\left(2t\right)=\mathrm{cos}\text{ }t$

39. $\mathrm{cos}\left(2t\right)=\mathrm{sin}\text{ }t$

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$\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$

40. $\mathrm{cos}\left(6x\right)-\mathrm{cos}\left(3x\right)=0$

For the following exercises, solve exactly on the interval $\text{ }\left[0,2\pi \right).\text{ }$ Use the quadratic formula if the equations do not factor.

41. ${\mathrm{tan}}^{2}x-\sqrt{3}\text{ }\mathrm{tan}\text{ }x=0$

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$0,\frac{\pi }{3},\pi ,\frac{4\pi }{3}$

42. ${\mathrm{sin}}^{2}x+\mathrm{sin}\text{ }x-2=0$

43. ${\mathrm{sin}}^{2}x-2\text{ }\mathrm{sin}\text{ }x-4=0$

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44. $5\text{ }{\mathrm{cos}}^{2}x+3\text{ }\mathrm{cos}\text{ }x-1=0$

45. $3\text{ }{\mathrm{cos}}^{2}x-2\text{ }\mathrm{cos}\text{ }x-2=0$

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${\mathrm{cos}}^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right),2\pi -{\mathrm{cos}}^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right)$

46. $5\text{ }{\mathrm{sin}}^{2}x+2\text{ }\mathrm{sin}\text{ }x-1=0$

47. ${\mathrm{tan}}^{2}x+5\mathrm{tan}\text{ }x-1=0$

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${\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),\pi +{\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right),\pi +{\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),2\pi +{\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right)$

48. ${\mathrm{cot}}^{2}x=-\mathrm{cot}\text{ }x$

49. $-{\mathrm{tan}}^{2}x-\mathrm{tan}\text{ }x-2=0$

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For the following exercises, find exact solutions on the interval $\text{ }\left[0,2\pi \right).\text{ }$ Look for opportunities to use trigonometric identities.

50. ${\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x-\mathrm{sin}\text{ }x=0$

51. ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=0$

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52. $\mathrm{sin}\left(2x\right)-\mathrm{sin}\text{ }x=0$

53. $\mathrm{cos}\left(2x\right)-\mathrm{cos}\text{ }x=0$

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$0,\frac{2\pi }{3},\frac{4\pi }{3}$

54. $\frac{2\text{ }\mathrm{tan}\text{ }x}{2-{\mathrm{sec}}^{2}x}-{\mathrm{sin}}^{2}x={\mathrm{cos}}^{2}x$

55. $1-\mathrm{cos}\left(2x\right)=1+\mathrm{cos}\left(2x\right)$

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$\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

56. ${\mathrm{sec}}^{2}x=7$

57. $10\text{ }\mathrm{sin}\text{ }x\text{ }\mathrm{cos}\text{ }x=6\text{ }\mathrm{cos}\text{ }x$

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${\mathrm{sin}}^{-1}\left(\frac{3}{5}\right),\frac{\pi }{2},\pi -{\mathrm{sin}}^{-1}\left(\frac{3}{5}\right),\frac{3\pi }{2}$

58. $-3\text{ }\mathrm{sin}\text{ }t=15\text{ }\mathrm{cos}\text{ }t\text{ }\mathrm{sin}\text{ }t$

59. $4\text{ }{\mathrm{cos}}^{2}x-4=15\text{ }\mathrm{cos}\text{ }x$

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${\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right)$

60. $8\text{ }{\mathrm{sin}}^{2}x+6\text{ }\mathrm{sin}\text{ }x+1=0$

61. $8\text{ }{\mathrm{cos}}^{2}\theta =3-2\text{ }\mathrm{cos}\text{ }\theta$

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$\frac{\pi }{3},{\mathrm{cos}}^{-1}\left(-\frac{3}{4}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{3}{4}\right),\frac{5\pi }{3}$

62. $6\text{ }{\mathrm{cos}}^{2}x+7\text{ }\mathrm{sin}\text{ }x-8=0$

63. $12\text{ }{\mathrm{sin}}^{2}t+\mathrm{cos}\text{ }t-6=0$

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${\mathrm{cos}}^{-1}\left(\frac{3}{4}\right),{\mathrm{cos}}^{-1}\left(-\frac{2}{3}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{2}{3}\right),2\pi -{\mathrm{cos}}^{-1}\left(\frac{3}{4}\right)$

64. $\mathrm{tan}\text{ }x=3\text{ }\mathrm{sin}\text{ }x$

65. ${\mathrm{cos}}^{3}t=\mathrm{cos}\text{ }t$

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$0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$

Graphical

For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.

66. $6\text{ }{\mathrm{sin}}^{2}x-5\text{ }\mathrm{sin}\text{ }x+1=0$

67. $8\text{ }{\mathrm{cos}}^{2}x-2\text{ }\mathrm{cos}\text{ }x-1=0$

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$\frac{\pi }{3},{\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right),\frac{5\pi }{3}$

68. $100\text{ }{\mathrm{tan}}^{2}x+20\text{ }\mathrm{tan}\text{ }x-3=0$

69. $2\text{ }{\mathrm{cos}}^{2}x-\mathrm{cos}\text{ }x+15=0$

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70. $20\text{ }{\mathrm{sin}}^{2}x-27\text{ }\mathrm{sin}\text{ }x+7=0$

71. $2\text{ }{\mathrm{tan}}^{2}x+7\text{ }\mathrm{tan}\text{ }x+6=0$

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$\pi +{\mathrm{tan}}^{-1}\left(-2\right),\pi +{\mathrm{tan}}^{-1}\left(-\frac{3}{2}\right),2\pi +{\mathrm{tan}}^{-1}\left(-2\right),2\pi +{\mathrm{tan}}^{-1}\left(-\frac{3}{2}\right)$

72. $130\text{ }{\mathrm{tan}}^{2}x+69\text{ }\mathrm{tan}\text{ }x-130=0$

Technology

For the following exercises, use a calculator to find all solutions to four decimal places.

73. $\mathrm{sin}\text{ }x=0.27$

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$2\pi k+0.2734,2\pi k+2.8682$

74. $\mathrm{sin}\text{ }x=-0.55$

75. $\mathrm{tan}\text{ }x=-0.34$

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$\pi k-0.3277$

76. $\mathrm{cos}\text{ }x=0.71$

For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval $\text{ }\left[0,2\pi \right).\text{ }$ Round to four decimal places.

77. ${\mathrm{tan}}^{2}x+3\text{ }\mathrm{tan}\text{ }x-3=0$

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$0.6694,1.8287,3.8110,4.9703$

78. $6\text{ }{\mathrm{tan}}^{2}x+13\text{ }\mathrm{tan}\text{ }x=-6$

79. ${\mathrm{tan}}^{2}x-\mathrm{sec}\text{ }x=1$

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$1.0472,3.1416,5.2360$

80. ${\mathrm{sin}}^{2}x-2\text{ }{\mathrm{cos}}^{2}x=0$

81. $2\text{ }{\mathrm{tan}}^{2}x+9\text{ }\mathrm{tan}\text{ }x-6=0$

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$0.5326,1.7648,3.6742,4.9064$

82. $4\text{ }{\mathrm{sin}}^{2}x+\mathrm{sin}\left(2x\right)\mathrm{sec}\text{ }x-3=0$

Extensions

For the following exercises, find all solutions exactly to the equations on the interval $\text{ }\left[0,2\pi \right).$

83. ${\mathrm{csc}}^{2}x-3\text{ }\mathrm{csc}\text{ }x-4=0$

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${\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\pi -{\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\frac{3\pi }{2}$

84. ${\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x-1=0$

85. ${\mathrm{sin}}^{2}x\left(1-{\mathrm{sin}}^{2}x\right)+{\mathrm{cos}}^{2}x\left(1-{\mathrm{sin}}^{2}x\right)=0$

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

86. $3\text{ }{\mathrm{sec}}^{2}x+2+{\mathrm{sin}}^{2}x-{\mathrm{tan}}^{2}x+{\mathrm{cos}}^{2}x=0$

87. ${\mathrm{sin}}^{2}x-1+2\text{ }\mathrm{cos}\left(2x\right)-{\mathrm{cos}}^{2}x=1$

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88. ${\mathrm{tan}}^{2}x-1-{\mathrm{sec}}^{3}x\text{ }\mathrm{cos}\text{ }x=0$

89. $\frac{\mathrm{sin}\left(2x\right)}{{\mathrm{sec}}^{2}x}=0$

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$0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$

90. $\frac{\mathrm{sin}\left(2x\right)}{2{\mathrm{csc}}^{2}x}=0$

91. $2\text{ }{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-\mathrm{cos}\text{ }x-5=0$

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92. $\frac{1}{{\mathrm{sec}}^{2}x}+2+{\mathrm{sin}}^{2}x+4\text{ }{\mathrm{cos}}^{2}x=4$

Real-World Applications

93. An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?

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${7.2}^{\circ }$

94. If a loading ramp is placed next to a truck, at a height of 4 feet, and the ramp is 15 feet long, what angle does the ramp make with the ground?

95. If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?

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${5.7}^{\circ }$

96. A woman is watching a launched rocket currently 11 miles in altitude. If she is standing 4 miles from the launch pad, at what angle is she looking up from horizontal?

97. An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is she looking down at him from horizontal? (Hint: this is called the angle of depression.)

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${82.4}^{\circ }$

98. A woman is standing 8 meters away from a 10-meter tall building. At what angle is she looking to the top of the building?

99. A man is standing 10 meters away from a 6-meter tall building. Someone at the top of the building is looking down at him. At what angle is the person looking at him?

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${31.0}^{\circ }$

100. A 20-foot tall building has a shadow that is 55 feet long. What is the angle of elevation of the sun?

101. A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?

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${88.7}^{\circ }$

102. A spotlight on the ground 3 meters from a 2-meter tall man casts a 6 meter shadow on a wall 6 meters from the man. At what angle is the light?

103. A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?

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${59.0}^{\circ }$

For the following exercises, find a solution to the word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.

104. A person does a handstand with his feet touching a wall and his hands 1.5 feet away from the wall. If the person is 6 feet tall, what angle do his feet make with the wall?

105. A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?

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${36.9}^{\circ }$

106. A 23-foot ladder is positioned next to a house. If the ladder slips at 7 feet from the house when there is not enough traction, what angle should the ladder make with the ground to avoid slipping?

Trigonometric Functions