5.2E: Solving Trigonometric Equations with Sine and Cosine (Exercises)
- Page ID
- 99746
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Part 1: Sine and cosine equations with common angles
In exercises # 1 - 6, find the exact value of all solutions to the equations on the interval \(0\le \theta <2\pi\). Give your solutions in radians.
1. \(\sin \left(\theta \right)=\dfrac{\sqrt{3}}{2} \)
2. \(\cos \left(\theta \right)=\dfrac{-\sqrt{2}}{2}\)
3. \(2\cos \left(\theta \right)-1=0\)
4. \(2\sin \left(\theta \right)+1=0\)
5. \(\sin \left(\theta \right)=1\)
6. \(\cos \left(\theta \right)=0\)
In exercises # 7 - 10, find the exact value of all solutions to the equations. Give your solutions in radians.
7. \(2\cos \left(\theta \right)-\sqrt{3}=0\)
8. \(2\sin \left(\theta \right) +\sqrt{3}=0\)
9. \(\sin \left(\theta \right)=0\)
10. \(\cos \left(\theta \right)=-1\)
Part 2: Sine and cosine equations with any angle
11. If \(\sin ^{-1} \left(a \right) = b\), write a corresponding expression with sine.
12. If \(\cos \left(u \right) = w\), write a corresponding expression with cosine inverse.
In exercises #13 - 16, evaluate the following expressions without a calculator. Express the answer in radians.
13. \(\sin ^{-1} \left(\dfrac{\sqrt{2} }{2} \right)\)
14. \(\sin ^{-1} \left(-\dfrac{1}{2} \right)\)
15. \(\cos ^{-1} \left(\dfrac{1}{2} \right)\)
16. \(\cos ^{-1} \left(-\dfrac{\sqrt{3} }{2} \right)\)
In exercises # 17 - 21, find all solutions to the equations. Express your results in degrees.
17. \(\cos \left(\theta \right)=0.064\)
18. \(\sin \left(\theta \right) = 0.915\)
19. \(3\sin \left(\theta \right) + 1 = 0\)
20. \(5\cos \left(\theta \right) +2 = 0\)
21. \(8\sin \left(\theta \right) -3 = \sin \left(\theta \right)\)
Part 3: Sine and cosine equations with multiple angles
In exercises #22 - 27, find all solutions to the equations. Express your results in degrees.
22. \(\sin \left(3\theta \right)=\dfrac{\sqrt{3}}{2}\)
23. \(\cos \left(2\theta \right)=0.27\)
24. \(2\sin \left(4\theta \right) +\sqrt{2} = 0\)
25. \(3\cos \left(\dfrac{\theta}{2} \right) - 1= 0\)
26. \(5\sin \left(6\theta \right)=0.83\)
27. \(2\cos \left(2\theta \right) + 2.1 =0.23\)
In exercises #28 - 30, find the four smallest positive solutions to the equations. Express your results in degrees.
28. \(2\sin \left(5\theta \right)+1=0\)
29. \(4\cos \left(8\theta \right) +3=0\)
30. \(2.5\sin \left(3.2\theta \right)-3.5=-2.1\)
31. The height of the water in a bay changes over time with the tides. Suppose that the height of the water in feet \( t \) hours since midnight is modeled by the formula \(h(t)= 3.8\sin \left(0.73t \right)+6 \).
- What is the height of the water at 6 am?
- When will the tide reach 8 ft. in height?
- When will the tide reach 10 ft. in height?
32. The volume of air in a person's lungs oscillates up and down as they breath. Suppose the volume of air in a person's lungs in Liters after \(t\) seconds is modeled by the formula \(V(t)= 4.21\cos \left(2.51t \right)+5.07 \).
- What is the volume of air in the person's lungs after 3 seconds?
- When will the the volume of air reach 8 Liters ?
33. Explain the mistake(s) that is(are) made.
Find the exact value of all solutions to the equation on the interval \(30\le \theta < 360\mathrm{{}^\circ} \).
\[\sin \left(\theta \right)=\dfrac{1}{2} \nonumber\]
Step 1: One solution is 30\(\mathrm{{}^\circ}\) since \(\sin \left(30\mathrm{{}^\circ} \right)=\dfrac{1}{2}\)
Step 2: The second solution is in quadrant two since sine and \(y\) are positive in quadrant two.
Step 3: The second solution can be found using the reference angle 0\(\mathrm{{}^\circ}\). So \(\theta = 90\mathrm{{}^\circ}+30\mathrm{{}^\circ}= 120\mathrm{{}^\circ}\)
- Answer
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3. \(\theta = \dfrac{\pi}{3}\) and \(\theta = \dfrac{5\pi}{3}\)
7. \(\theta = \dfrac{\pi}{6} + 2\pi n\) and \(\theta = \dfrac{-\pi}{6} + 2\pi n\) where \(n\) is any integer.
13. \(\theta = \dfrac{\pi}{4} \)
17. \(\theta \approx 86.33^{\circ} + 360^{\circ}n\) and \(\theta \approx -86.33^{\circ} + 360^{\circ}n\) where \(n\) is any integer.
22. \(\theta = 20^{\circ} + 120^{\circ}n\) and \(\theta = 40^{\circ} + 120^{\circ}n\) where \(n\) is any integer.
28. \(\theta = 42^{\circ}\), \(\theta = 66^{\circ}\), \(\theta = 114^{\circ}\), and \(\theta = 138^{\circ}\)