3.4.1: Exercises 3.4
- Page ID
- 63466
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
Explain why the domain of tangent is restricted.
- Answer
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Tangent is undefined whenever cosine is 0.
Exercise \(\PageIndex{2}\)
Explain why the domain of cosecant is restricted.
- Answer
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Cosecant is undefined whenever sine is 0.
Exercise \(\PageIndex{3}\)
Explain what is meant by the range of a function.
- Answer
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The range describes the possible output values of the function.
Exercise \(\PageIndex{4}\)
What do the coordinates on the unit circle tell you?
- Answer
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The x coordinate tells you the value of cosine for that angle and the y coordinate tells you the value of sine for that angle.
Exercise \(\PageIndex{5}\)
Sketch the unit circle from memory. Use Figure 3.4.2 to check your work and add in any values you could not remember.
- Answer
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Problems
Evaluate each statement given in exercises \(\PageIndex{6}\) - \(\PageIndex{10}\).
Exercise \(\PageIndex{6}\)
\(\displaystyle \tan{\bigg( \frac{\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle 1\)
Exercise \(\PageIndex{7}\)
\(\displaystyle \cos{\bigg( \frac{-\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle \frac{\sqrt{2}}{2}\)
Exercise \(\PageIndex{8}\)
\(\displaystyle \sin{\bigg( \frac{3\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle \frac{\sqrt{2}}{2}\)
Exercise \(\PageIndex{9}\)
\(\displaystyle \csc{\bigg( \frac{-3\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle -\sqrt{2}\)
Exercise \(\PageIndex{10}\)
\(\displaystyle \sin{\bigg( \frac{3\pi}{2}\bigg)}\)
- Answer
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\(\displaystyle -1\)
Determine the range of each function given in exercises \(\PageIndex{11}\) - \(\PageIndex{14}\).
Exercise \(\PageIndex{11}\)
\(f(x) = -2 \sin{(4x)} + 3\)
- Answer
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\([1,5]\)
Exercise \(\PageIndex{12}\)
\(g(x) = 6\cos{(2x)} -8\)
- Answer
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\([-14,-2]\)
Exercise \(\PageIndex{13}\)
\(h(x) = -\sin{(x)} -1\)
- Answer
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\([-2,0]\)
Exercise \(\PageIndex{14}\)
\(f(\theta) =4\sin{(\theta-\pi)}\)
- Answer
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\([-4,4]\)
In exercises \(\PageIndex{15}\) - \(\PageIndex{18}\), use the unit circle to help you answer the given question.
Exercise \(\PageIndex{15}\)
Find the ordered pair for the point on the unit circle associated with \(\theta=\frac{5\pi}{4}\)
- Answer
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\((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\)
Exercise \(\PageIndex{16}\)
Sketch the a unit circle and the angle represented by \(\theta = \frac{7\pi}{6}\). Find the ordered pair where this line intersects the unit circle and label this point on your sketch.
- Answer
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Exercise \(\PageIndex{17}\)
Sketch the a unit circle and the angle represented by \(\theta = -\frac{2\pi}{3}\). Find the ordered pair where this line intersects the unit circle and label this point on your sketch.
- Answer
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Exercise \(\PageIndex{18}\)
Find the equation of the line that intersects the unit circle at \(\theta = \pi\) and at \(\theta=\frac{\pi}{3}\). Answer in slope intercept form.
- Answer
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\(y=\frac{\sqrt{3}}{3} x+\frac{\sqrt{3}}{3}\)