In Exercises othergraphsfirst - othergraphslast, graph one cycle of the given function. State the period of the function.

\(y = \tan \left (t - \frac {\pi }{3} \right )\)

Use the Desmos graph below with the settings \(f(t)=\tan (t),\,h=\frac {\pi }{3},\,b=1,\,a=1,\,k=0\)

Period: \(\pi \)

\(y = 2\tan \left ( \frac {1}{4}t \right ) - 3\)

Use the Desmos graph below with the settings \(f(t)=\tan (t),\,h=0,\,b=\frac {1}{4},\,a=2,\,k=-3\)

Period: \(4\pi \)

\(y = \frac {1}{3}\tan (-2t - \pi ) + 1\)

Use the Desmos graph below with the settings \(f(t)=\tan (t),\,h=\pi ,\,b=-2,\,a=\frac {1}{3},\,k=1\)

Period: \(\frac {\pi }{2}\)

\(y = \sec \left ( t - \frac {\pi }{2} \right )\)

Use the Desmos graph below with the settings \(f(t)=\sec (t),\,h=\frac {\pi }{2},\,b=1,\,a=1,\,k=0\)

To graph by hand, start with \(y = \cos \left ( t - \frac {\pi }{2} \right )\)
Period: \(2\pi \)

\(y = -\csc \left ( t + \frac {\pi }{3} \right )\)

Use the Desmos graph below with the settings \(f(t)=\csc (t),\,h=-\frac {\pi }{3},\,b=\frac {1}{2},\,a=\frac {1}{3},\,k=0\)

To graph by hand, start with \(y = -\frac {1}{3}\cos \left ( \frac {1}{2}t + \frac {\pi }{3} \right )\)
Period: \(4\pi \)

\(y = -\frac {1}{3} \sec \left ( \frac {1}{2}t + \frac {\pi }{3} \right )\)

Use the Desmos graph below with the settings \(f(t)=\sec (t),\,h=-\frac {\pi }{3},\,b=1,\,a=-\frac {1}{3},\,k=0\)

To graph by hand, start with \(y = -\frac {1}{3}\cos \left ( \frac {1}{2}t + \frac {\pi }{3} \right )\)
Period: \(4\pi \)

\(y = \csc (2t - \pi )\)

Use the Desmos graph below with the settings \(f(t)=\csc (t),\,h=\pi ,\,b=2,\,a=1,\,k=0\)

To graph by hand, start with \(y = \sin (2t - \pi )\)
Period: \(\pi \)

\(y = \sec (3t - 2\pi ) + 4\)

Use the Desmos graph below with the settings \(f(t)=\sec (t),\,h=2\pi ,\,b=3,\,a=1,\,k=4\)

To graph by hand, start with \(y = \cos (3t - 2\pi ) + 4\)
Period: \(\frac {2\pi }{3}\)

\(y = \csc \left ( -t - \frac {\pi }{4} \right ) - 2\)

Use the Desmos graph below with the settings \(f(t)=\csc (t),\,h=\frac {\pi }{4},\,b=-1,\,a=1,\,k=-2\)

To graph by hand, start with \(y = \sin \left ( -t - \frac {\pi }{4} \right ) - 2\)
Period: \(2\pi \)

\(y = \cot \left ( t + \frac {\pi }{6} \right )\)

Use the Desmos graph below with the settings \(f(t)=\cot (t),\,h=-\frac {\pi }{6},\,b=1,\,a=1,\,k=0\)

Period: \(\pi \)

\(y = -11\cot \left ( \frac {1}{5} t \right )\)

Use the Desmos graph below with the settings \(f(t)=\cot (t),\,h=-11,\,b=\frac {1}{5},\,a=1,\,k=0\)

Period: \(5\pi \)

\(y = \frac {1}{3} \cot \left ( 2t + \frac {3\pi }{2} \right ) + 1\)

Use the Desmos graph below with the settings \(f(t)=\cot (t),\,h=-\frac {3\pi }{2},\,b=2,\,a=\frac {1}{3},\,k=1\)

Period: \(\frac {\pi }{2}\)

In Exercises fitsecantcosecantfirst - fitsecantcosecantlast, the graph of a (co)secant function is given. Find a formula for the function in the form \(F(t) = A \sec (\omega t + \phi ) + B\) and \(G(t) = A \csc (\omega t + \phi ) + B\). Select \(\omega \) so \(\omega > 0\). Check your answer by graphing.

Asymptotes: \(t = \pm \frac {\pi }{2}\), \(t=\pm \frac {3\pi }{2}\), …

[Picture]

\(F(t) = 2 \sec (t-\pi )\), \(G(t) = 2 \csc \left (t - \frac {\pi }{2} \right )\)

Use the Desmos graph below to check that both of your formulas agree.

Asymptotes: \(t = \pm 1\), \(t = \pm 3\), \(t = \pm 5\), …

[Picture]

\(F(t) = \sec \left ( \frac {\pi }{2} t \right ) + 1\), \(G(t) = \csc \left ( \frac {\pi }{2} t + \frac {\pi }{2} \right ) + 1\)

Use the Desmos graph below to check that both of your formulas agree.

In Exercises fittangentfirst - fittangentlast, the graph of a (co)tangent function given. Find a formula the function in the form \(J(t) = A \tan (\omega t + \phi ) + B\) and \(K(t) = A \cot (\omega t + \phi ) + B\). Select \(\omega \) so \(\omega > 0\). Check your answer by graphing.

Asymptotes: \(t=-\frac {3 \pi }{4}\), \(t=\frac {\pi }{4}\), \(t = \frac {5\pi }{4}\), …

[Picture]

\(J(t) = -\tan \left (t+ \frac {\pi }{4} \right )\), \(K(t) = \cot \left (t - \frac {\pi }{4} \right )\)

Use the Desmos graph below to check that both of your formulas agree.

Asymptotes: \(t = \pm 2\), \(t = \pm 6\), \(t = \pm 10\), …

[Picture]

\(J(t) = \tan \left ( \frac {\pi }{4} t \right ) + 1\), \(K(t) = -\cot \left ( \frac {\pi }{4} t + \frac {\pi }{2} \right ) + 1\)

Use the Desmos graph below to check that both of your formulas agree.

In Section SineCosineLimits, we observed that the cosine and sine functions are continuous. As such, the other four circular functions are continuous on their domains. In Exercises othertriglimitexfirst - othertriglimitexlast, determine the given limit. Use the symbols ‘\(-\infty \)’ and ‘\(\infty \)’as appropriate. Check your answers graphically.
\(\ds {\lim _{t \rightarrow 0} \tan (t)}\).

\(\ds {\lim _{t \rightarrow 0} \tan (t) = \tan (0) = 0}\).
\(\ds {\lim _{t \rightarrow \pi } \sec (t)}\)

\(\ds {\lim _{t \rightarrow \pi } \sec (t) = \sec (\pi ) = -1}\)
\(\ds {\lim _{t \rightarrow 3\pi }}\) \(\cot \left (\frac {t}{2}\right )\)

\(\ds {\lim _{t \rightarrow 3\pi }}\) \(\cot \left (\frac {t}{2}\right ) = \cot \left (\frac {3\pi }{2}\right ) = 0\)
\(\ds {\lim _{\theta \rightarrow 0}}\) \(\csc \left (2\theta + \frac {\pi }{4}\right )\).

\(\ds {\lim _{\theta \rightarrow 0}}\) \(\csc \left (2\theta + \frac {\pi }{4}\right ) = \csc \left (\frac {\pi }{4}\right ) = \sqrt {2}\).
\(\ds {\lim _{\theta \rightarrow \pi } (\cos (\theta ) - \sec (\theta ))}\)

\(\ds {\lim _{\theta \rightarrow \pi } (\cos (\theta ) - \sec (\theta )) = \cos (\pi ) - \sec (\pi ) = -1 - (-1) = 0}\)
\(\ds {\lim _{\theta \rightarrow \frac {\pi }{4}} \sec (\theta ) \, \tan (\theta )}\)

\(\ds {\lim _{\theta \rightarrow \frac {\pi }{4}} (\sec (\theta ) \, \tan (\theta )) = \sec \left ( \frac {\pi }{4} \right ) \tan \left ( \frac {\pi }{4} \right ) = \sqrt {2}}\)
\(\ds {\lim _{x \rightarrow 0^{+}} \csc (3x)}\)

\(\ds {\lim _{x \rightarrow 0^{+}} \csc (3x) = \infty }\)
\(\ds {\lim _{x \rightarrow \pi ^{-}} (\sin (2x) - \tan (x))}\)

\(\ds {\lim _{x \rightarrow \pi ^{-}} (\sin (2x) - \tan (x))} = \sin (2\pi ) - \tan (\pi ) = 0\)
\(\ds {\lim _{x \rightarrow \pi ^{+}}}\) \((\cos (x) + \cot (x))\)

\(\ds {\lim _{x \rightarrow \pi ^{+}}}\) \((\cos (2x) + \cot (x)) = \infty \)

In Exercise sintovertexercise3 in Section TheOtherCircularFunctions, we proved \(\ds {\lim _{\theta \rightarrow 0}}\) \(\frac {\sin (\theta )}{\theta } = 1\).

  1. Use Theorem LimitProp01 from Section IntroLimits to show \(\ds {\lim _{\theta \rightarrow 0}}\) \(\frac {\theta }{\sin (\theta )} = 1\).

    \(\ds {\lim _{\theta \rightarrow 0}}\) \(\frac {\theta }{\sin (\theta )}\) \( = \ds {\lim _{\theta \rightarrow 0}}\) \(\left [\frac {\sin (\theta )}{\theta }\right ]^{-1} = 1^{-1} = 1\)

  2. Which indeterminate form is present in the limit \(\ds {\lim _{\theta \rightarrow 0^{+}} 2 \theta \, \csc (\theta )}\)?

    As \(\theta \rightarrow 0^{+}\), \(2 \theta \, \csc (\theta ) \rightarrow 0 \cdot \infty \).

  3. Graph \(f(\theta ) = 2 \theta \, \csc (\theta )\) near \(\theta = 0\). What appears to be the limit?

    The graph of \(f(\theta ) = 2 \theta \, \csc (\theta )\) approaches \((0,2)\) so \(\ds {\lim _{\theta \rightarrow 0^{+}} 2 \theta \, \csc (\theta )}\) appears to be \(2\).

  4. Rewrite \(2 \theta \, \csc (\theta )\) in terms of \(\theta \) and \(\sin (\theta )\) and use part thetaoversinetheta to analytically find \(\ds {\lim _{\theta \rightarrow 0^{+}} 2 \theta \, \csc (\theta )}\).

    \(\ds {\lim _{\theta \rightarrow 0^{+}} 2 \theta \, \csc (\theta )}\) \(= \ds {\lim _{\theta \rightarrow 0^{+}}}\) \(2 \, \theta \frac {1}{\sin (\theta )}\) \(= \ds {\lim _{\theta \rightarrow 0^{+}}}\) \(2 \, \frac {\theta }{\sin (\theta )} = 2(1) = 2\).

  5. Use the same methodology as above to help you analytically determine \(\ds {\lim _{\theta \rightarrow 0^{+}} 2 \theta \, \cot (\theta )}\)

    \(\ds {\lim _{\theta \rightarrow 0^{+}} 2 \theta \, \cot (\theta )}\) \(= \ds {\lim _{\theta \rightarrow 0^{+}}}\) \(2 \, \theta \, \frac {\cos (\theta )}{\sin (\theta )}\) \(= \ds {\lim _{\theta \rightarrow 0^{+}}}\) \(2 \, \cos (\theta ) \, \frac {\theta }{\sin (\theta )} = 2\cos (0)(1) = 2(1)(1) = 2\)
Use the conversion formulas listed in Theorem cosinesinefunctionprops to create conversion formulas between secant and cosecant functions.

\(\csc \left (t + \frac {\pi }{2} \right ) = \sec (t)\) and \(\sec \left (t - \frac {\pi }{2} \right ) = \csc (t)\).

\(f(t) = \sec \left ( 2 t - \frac {7\pi }{6} \right ) -1 = \csc \left ( \left [2 t - \frac {7\pi }{6}\right ] + \frac {\pi }{2} \right ) -1 = \csc \left ( 2 t - \frac {2\pi }{3} \right ) -1 \), in terms of cosecants.
Use a conversion formula to rewrite our first answer to Example secantcosecantfromgraphex, \(f(t) = \sec \left ( 2 t - \frac {7\pi }{6} \right ) -1\), in terms of cosecants.
Rework Example secantcosecantfromgraphex and find answers with \(A<0\).
Prove Theorem secantcosecanttperiodphaseshift using Theorem sinusoidform.

In this Exercise, we argue the range of the tangent function is \((-\infty , \infty )\). Let \(M\) be a fixed, but arbitrary positive real number.

  1. Show there is an acute angle \(\theta \) with \(\tan (\theta ) = M\). (Hint: think right triangles.)
  2. Using the symmetry of the Unit Circle, explain why there are angles \(\theta \) with \(\tan (\theta ) = -M\).
  3. Find angles with \(\tan (\theta ) = 0\).
  4. Combine the three parts above to conclude the range of the tangent function is \((-\infty , \infty )\).
Prove \(\cot (t)\) is odd. (Hint: mimic the proof given in the text that \(\tan (t)\) is odd.)