5.6E: Simpliying and Solving Equations with Trigonometric Functions (Exercises)
- Page ID
- 99754
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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: The reciprocal and quotient identities
In exercises # 1 - 10, perform the indicated operation and simplify the expression. Write the expression without fraction and/or as a single trig term if possible.
1. \(\cos (t)\tan(t)\)
2. \(\csc (t)\tan(t)\)
3. \(\dfrac{\sec(t)}{\csc(t)}\)
4. \(\dfrac{\cot(t)}{\csc(t)}\)
5. \( \sin^2(\theta)(\csc^2(\theta)-1) \)
6. \(\csc(t) (\cos(t)+\sin(t) )\)
7. \(\sec(t) - \tan(t)\sin(t)\)
8. \(\dfrac{\cos(t)}{\sin(t) \cot(t)}\)
9. \(\cos(t) (\sec(t) + \csc(t))\)
10. \( (\sec(\theta)+\csc(\theta))(\cos(\theta) - \sin(\theta)) \)
In exercises #11 - 17, find all solutions to the equations
11. \( \csc(\theta) = 4 \)
12. \(3 \sec(\theta) + 2 = 5 \)
13. \(4\cot^2(\theta)-1 = 0 \)
14. \(2\tan(\theta)\cos(\theta)-1=0 \)
15. \(\sin(\theta) - 2\csc(\theta) = 0 \)
16. \(\sin(\theta)\cos(\theta)=0 \)
17. \(2 \sin( \theta) - 1 = \csc( \theta) \)
18. \(3 \sec^2(\theta) + \tan^2(\theta) = 0 \)
Part 2: Pythagorean Identities
In exercises # 19-25, perform the indicated operation and simplify the expression. Write the expression without fraction if possible.
19. \(\cos^2(\theta)+ \cos^2(\theta)\tan^2(\theta) \)
20. \( (1-\cos(\theta))(1+\cos(\theta)) \)
21. \( \sqrt{25-25cos^2(\theta)} \)
22. \(\dfrac{\sin ^{2} \left(t\right)+\cos ^{2} \left(t\right)}{\cos ^{2} \left(t\right)}\)
23. \(\dfrac{1-\sin ^{2} \left(t\right)}{\sin ^{2} \left(t\right)}\)
24. \( \dfrac{1}{\text{cos}^2 (t)} - 1\)
25. \(\dfrac{\cos(t)}{\sec(t)}-\dfrac{\sin(t)}{\csc(t)}\)
In exercises #26 - 30, find all solutions to the equation
26. \(2 \cos^2(\theta)+ \sin^2(\theta)-1=0 \)
27. \(3 \sec^2(\theta) + \tan^2(\theta) = 1 \)
28. \( 2 \sin(\theta) - 1 = \csc(\theta) \)
- Answer
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1. \( \sin(t) \)
6. \( 1+ \tan(t) \)
11. \(\theta \approx 14.48^{\circ} + 360^{\circ}n\) and \(\theta \approx 165.52^{\circ} + 360^{\circ}n\) where \(n\) is any integer.
16. \(\theta = 90^{\circ}n\) where \(n\) is any integer.
19. \( 1\)
26. \(\theta = 90^{\circ} + 360^{\circ}n\) and \(\theta \approx 270^{\circ} + 360^{\circ}n\) where \(n\) is any integer.