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In Exercises whereisanglefirst - whereisanglelast, use the given the information to determine the quadrant in which the terminal side of the angle lies when
plotted in standard position.
In Exercises circcalcfirst - circcalclast, use your calculator to approximate the given value to three decimal places. Make sure your calculator is in
the proper angle measurement mode!
For \(f(t) = 3t^2 + 2 \tan (3t)\), find functions \(g\) and \(h\) so that \(f=g+h\).
One solution is \(g(t) = 3t^2\) and \(h(t) = 2\tan (3t)\).
For \(f(\theta ) = \sec (\theta ) - \tan (\theta )\), find functions \(g\) and \(h\) so that \(f=g-h\).
One solution is \(g(\theta ) = \sec (\theta )\) and \(h(\theta ) = \tan (\theta )\).
For \(f(t) = -\csc (t) \cot (t)\), find functions \(g\) and \(h\) so that \(f=gh\).
One solution is \(g(t) = -\csc (t)\) and \(h(t) = \cot (t)\).
For \(r(t) = \frac {\tan (3t)}{t}\), find functions \(f\) and \(g\) so \(r = \frac {f}{g}\).
One solution is \(f(t) = \tan (3t)\) and \(g(t) = t\).
For \(T(\theta ) =\tan (4 \theta )\), find functions \(f\) and \(g\) so \(T = g \circ f\).
One solution is \(f(\theta ) = 4 \theta \) and \(g(\theta ) = \tan (\theta )\).
For \(s(\theta ) = \sec ^{2}(\theta )\), find functions \(f\) and \(g\) so \(s = g \circ f\).
Since \(\sec ^{2}(\theta ) = (\sec (\theta ))^2\), one solution is \(f(\theta ) = \sec (\theta )\) and \(g(\theta ) = \theta ^2\).
For \(L(x) = \ln (\sin (x) )\), find functions \(f\) and \(g\) so \(L = g \circ f\).
One solution is \(f(x) = \sin (x)\) and \(g(x) = \ln (x)\).
For \(\ell (\theta ) = \ln | \sec (\theta ) - \tan (\theta )|\), find find functions \(f\), \(g\), and \(h\) so \(\ell = h \circ (f-g)\).
One solution is \(f(\theta ) = \sec (\theta )\), \(g(\theta ) = \tan (\theta )\), and \(h(\theta ) = \ln | \theta |\).
Let \(S(t) = \sin (t)\) and \(C(t) = \cos (t)\), \(F(t) = \tan (t)\), and \(G(t) = \cot (t)\). Explain why \(F = \frac {S}{C}\) but \(F \neq \frac {1}{G}\).
HINT: Think about domains …
For each function \(T(t)\) listed below, compute the average rate of change over the indicated interval. (See Definition arc in
Section AverageRateofChange for a review of this concept, as needed.) What trends do you notice? Compare your answer with what you
discovered in Section TheCircularFunctionsSineandCosine number sinearcexercise. Be sure your calculator is in radian mode!
As we zoom in towards \(0\), the average rate of change of \(\tan (k t)\) approaches \(k\). This is the same trend we observed for \(\sin (k t)\) in Section TheCircularFunctionsSineandCosine
number sinearcexercise.
We wish to establish the inequality \(\cos (\theta ) < \frac {\sin (\theta )}{\theta } < 1\) for \(0 < \theta < \frac {\pi }{2}.\) Use the diagram from the beginning of the section, partially reproduced below, to
answer the following.
Show that triangle \(OPB\) has area \(\frac {1}{2} \sin (\theta )\) and triangle \(OQB\) has area \(\frac {1}{2} \tan (\theta )\).
Show that the circular sector \(OPB\) with central angle \(\theta \) has area \(\frac {1}{2} \theta \).
Comparing areas, show that \(\sin (\theta ) < \theta < \tan (\theta )\) for \(0 < \theta < \frac {\pi }{2}.\)
Use the inequality \(\sin (\theta ) < \theta \) to show that \(\frac {\sin (\theta )}{\theta } < 1\) for \(0 < \theta < \frac {\pi }{2}.\)
Use the inequality \(\theta < \tan (\theta )\) to show that \(\cos (\theta ) < \frac {\sin (\theta )}{\theta }\) for \(0 < \theta < \frac {\pi }{2}.\) Combine this with the previous part to complete the proof.
Show that \(\cos (\theta ) < \frac {\sin (\theta )}{\theta } < 1\) also holds for \(-\frac {\pi }{2}< \theta < 0\).
Use the results from Exercises sintovertexercise1 and sintovertexercise2 along with the Squeeze Theorem, (Theorem squeezeth) to prove \(\ds { \lim _{\theta \rightarrow 0}}\) \( \frac {\sin (\theta )}{\theta } = 1\).
![[Picture]](ExercisesforTheOtherCircularFunctions.online-4bab5fcabbc6356c340d4d1e7e9831eb.svg)