Chapter 6

(a) Use a calculator to find the average rate of change of \(g(t)=\sin t\) from 2 to \(2+h,\) for each of these values of \(h: 01, .001, .0001,\) and .00001 (b) Compare your answers in part (a) with the number cos \(2 .\) What would you guess that the instantaneous rate of change of \(g(t)=\sin t\) is at \(t=2 ?\)

Graph the function. Does the function appear to be periodic? If so, what is the period? $$f(t)=|\cos t|$$

Weight is pulled 6 centimeters below equilibrium, and the initial movement is upward.

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\cos t=\frac{1}{2} \quad \text { and } \quad \sin t<0$$

Determine the positive radian measure of the angle that the second hand of a clock traces out in the given time. 1 minute and 55 seconds

(a) Use a calculator to find the average rate of change of \(f(t)=\cos t\) from 5 to \(5+h,\) for each of these values of \(h: 01, .001, .0001,\) and .00001 (b) Compare your answers in part (a) with the number \(-\sin 5 .\) What would you guess that the instantaneous rate of change of \(f(t)=\cos t\) is at \(t=5 ?\)

Graph the function. Does the function appear to be periodic? If so, what is the period? $$g(t)=\sin |t|$$

A pendulum swings uniformly back and forth, taking 2 seconds to move from the position directly above point \(A\) to the position directly above point \(B\). (Check your book to see image) The distance from \(A\) to \(B\) is 20 centimeters. Let \(d(t)\) be the horizontal distance from the pendulum to the (dashed) center line at time \(t\) seconds (with distances to the right of the line measured by positive numbers and distances to the left by negative ones). Assume that the pendulum is on the center line at time \(t=0\) and moving to the right. Assume that the motion of the pendulum is simple harmonic motion. Find the rule of the function \(d(t)\)

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\frac{1}{\cos t}-\sin t \tan t$$

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\cos t=0 \quad \text { and } \quad \sin t=1$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 3.5 minutes, 1 rpm

The diagram shows a merry-go-round that is turning counterclockwise at a constant rate, making 2 revolutions in 1 minute. On the merry-go-round are horses \(A, B, C,\) and \(D\) at 4 meters from the center and horses \(E, F,\) and \(G\) at 8 meters from the center. There is a function \(a(t)\) that gives the distance the horse \(A\) is from the \(y\) -axis (this is the \(x\) -coordinate of the position \(A\) is in ) as a function of time \(t\) (measured in minutes). Similarly, \(b(t)\) gives the \(x\) -coordinate for \(B\) as a function of time, and so on. Assume that the diagram shows the situation at time \(t=0\). (Check your book to see figure) (a) Which of the following functions does \(a(t)\) equal? $$\begin{array}{ll}4 \cos t, & 4 \cos \pi t, \quad 4 \cos 2 t, \quad 4 \cos 2 \pi t \\\4 \cos \left(\frac{1}{2} t\right), & 4 \cos ((\pi / 2) t), \quad 4 \cos 4 \pi t\end{array}$$ Explain. (b) Describe the functions \(b(t), c(t), d(t),\) and so on using the cosine function: $$\begin{array}{l}b(t)=\longrightarrow(t)=\longrightarrow d(t)= \\\e(t)=\longrightarrow f(t)=\longrightarrow g(t)=\end{array}$$ (c) Suppose the \(x\) -coordinate of a horse \(S\) is given by the function \(4 \cos (4 \pi t-(5 \pi / 6))\) and the \(x\) -coordinate of another horse \(T\) is given by \(8 \cos (4 \pi t-(\pi / 3))\) Where are these horses located in relation to the rest of the horses? Mark the positions of \(T\) and \(S\) at \(t=0\) into the figure.

Graph the function. Does the function appear to be periodic? If so, what is the period? $$g(t)=|\sin t|$$

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\frac{1-\tan ^{2} t}{1+\tan ^{2} t}+2 \sin ^{2} t$$

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\sin t=-2 / 3 \quad \text { and } \quad \sec t>0$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). \(t\) minutes, \(1 \mathrm{rpm}\)

In Exercises \(57-62,\) assume that the terminal side of an angle of t radians in standard position lies in the given quadrant on the given straight line. Find sin \(t,\) cos \(t,\) tan \(t .\) [Hint: Find \(a\) point on the terminal side of the angle.\(]\) Quadrant III; line with equation \(2 y-5 x=0\)

The average monthly temperature in Cleveland, Ohio is approximated by $$f(t)=22.7 \sin (.52 x-2.18)+49.6$$ where \(t=1\) corresponds to January, \(t=2\) to February, and so on. (a) Construct a table of values \((t=1,2, \ldots, 12)\) for the function \(f(t)\) and another table for \(f(t+12.083)\) (b) Based on these tables would you say that the function \(f\) is (approximately) periodic? If so, what is the period? Is this reasonable?

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\sec t=-13 / 5 \quad \text { and } \quad \tan t<0$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 1 minute, 2 rpm

Graph the function. Does the function appear to be periodic? If so, what is the period? $$h(t)=|\tan t|$$

A typical healthy person's blood pressure can be modeled by the periodic function $$f(t)=22 \cos (2.5 \pi t)+95$$ where \(t\) is time (in seconds) and \(f(t)\) is in millimeters of mercury. Which one of \(.5, .8,\) or 1 appears to be the period of this function?

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\csc t=8 \quad \text { and } \quad \cos t<0$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 3.5 minutes, \(2 \mathrm{rpm}\)

The table below shows the number of unemployed people in the labor force (in millions) for \(1984-2005 .\) (a) Sketch a scatter plot of the data, with \(x=0\) corresponding to 1980 (b) Does the data appear to be periodic? If so, find an appropriate model. (c) Do you think this model is likely to be accurate much beyond \(2005 ?\) Why? \(\begin{array}{|c|c|}\hline \text { Year } & \text { Unemployed } \\\\\hline 1984 & 8.539 \\\\\hline 1985 & 8.312 \\\\\hline 1986 & 8.237 \\\\\hline 1987 & 7.425 \\\\\hline 1988 & 6.701 \\\\\hline 1989 & 6.528 \\\\\hline 1990 & 7.047 \\\\\hline 1991 & 8.628 \\\\\hline 1992 & 9.613 \\\\\hline 1993 & 8.940 \\\\\hline 1994 & 7.996 \\\\\hline\end{array}\) \(\begin{array}{|c|c|}\hline \text { Year } & \text { Unemployed } \\\\\hline 1995 & 7.404 \\\\\hline 1996 & 7.236 \\\\\hline 1997 & 6.739 \\\\\hline 1998 & 6.210 \\\\\hline 1999 & 5.880 \\\\\hline 2000 & 5.692 \\\\\hline 2001 & 6.801 \\\\\hline 2002 & 8.378 \\\\\hline 2003 & 8.774 \\\\\hline 2004 & 8.149 \\\\\hline 2005 & 7.591 \\\\\hline\end{array}\)

Explore various ways in which a calculator can produce inaccurate graphs of trigonometric functions. These exercises also provide examples of two functions, with different graphs, whose graphs appear identical in certain viewing windows. Choose a viewing window with \(-3 \leq y \leq 3\) and \(0 \leq x \leq k\) where \(k\) is chosen as follows. $$\begin{array}{|l|c|} \hline \text { Width of Screen } & k \\ \hline \begin{array}{l} \text { 95 pixels } \\ (\mathrm{TI}-83 / 84+) \end{array} & 188 \pi \\ \hline \begin{array}{l} \text { 127 pixels } \\ \text { (TI-86, Casio) } \end{array} & 252 \pi \\ \hline \begin{array}{l} \text { 131 pixels } \\ \text { (HP-39gs) } \end{array} & 260 \pi \\ \hline \begin{array}{l} \text { 159 pixels } \\ \text { (TI-89) } \end{array} & 316 \pi \\ \hline \end{array}$$ (a) Graph \(y=\cos x\) and the constant function \(y=1\) on the same screen. Do the graphs look identical? Are the functions the same? (b) Use the trace feature to move the cursor along the graph of \(y=\cos x,\) starting at \(x=0 .\) For what values of \(x\) did the calculator plot points? [Hint: \(2 \pi \approx 6.28 .]\) Use this information to explain why the two graphs look identical.

The percentage of the face of the moon that is illuminated (as seen from earth) on day \(t\) of the lunar month is given by $$g(t)=.5\left(1-\cos \frac{2 \pi t}{29.5}\right)$$ (a) What percentage of the face of the moon is illuminated on day 0? Day 10? Day 22? (b) Construct appropriate tables to confirm that \(g\) is a periodic function with period 29.5 days. (c) When does a full moon occur \((g(t)=1) ?\)

In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\tan t=\cot \left(\frac{\pi}{2}-t\right)$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 4.25 minutes, \(5 \mathrm{rpm}\)

Do the following. (a) Use 12 data points (with \(x=1\) corresponding to January) to find a periodic model of the data. (b) What is the period of the function found in part ( \(a\) )? Is this reasonable? (c) Plot 24 data points (two years) and graph the function from part ( a ) on the same screen. Is the function a good model in the second year? (d) Use the 24 data points in part ( \(c\) ) to find another periodic model for the data. (e) What is the period of the function in part ( \(d\) )? Does its graph fit the data well? The table shows the average monthly temperature in Chicago, IIL, based on data from 1971 to 2000 . $$\begin{array}{|c|c|}\hline \text { Month } & \text { Temperature }\left(^{\circ} \mathrm{F}\right) \\\\\hline \text { Jan } & 22.0 \\\\\hline \text { Feb } & 27.0 \\\\\hline \text { Mar } & 37.3 \\\\\hline \text { Apr } & 47.8 \\\\\hline \text { May } & 58.7 \\\\\hline \text { Jun } & 68.2 \\\\\hline \text { Jul } & 73.3 \\\\\hline \text { Aug } & 71.7 \\\\\hline \text { Sep } & 63.8 \\\\\hline \text { Oct } & 52.1 \\\\\hline \text { Nov } & 39.3 \\\\\hline \text { Dec } & 27.4 \\\\\hline\end{array}$$

In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\cos t}{\cos (t-\pi / 2)}=\cot t$$

Do the following. (a) Use 12 data points (with \(x=1\) corresponding to January) to find a periodic model of the data. (b) What is the period of the function found in part ( \(a\) )? Is this reasonable? (c) Plot 24 data points (two years) and graph the function from part ( a ) on the same screen. Is the function a good model in the second year? (d) Use the 24 data points in part ( \(c\) ) to find another periodic model for the data. (e) What is the period of the function in part ( \(d\) )? Does its graph fit the data well? The table shows the average monthly precipitation (in inches) in San Francisco, CA, based on data from 1971 to 2000. $$\begin{array}{|c|c|}\hline \text { Month } & \text { Precipitation } \\\\\hline \text { Jan } & 4.45 \\\\\hline \text { Feb } & 4.01 \\\\\hline \text { Mar } & 3.26 \\\\\hline \text { Apr } & 1.17 \\\\\hline \text { May } & .38 \\\\\hline \text { Jun } & .11 \\\\\hline \text { Jul } & .03 \\\\\hline \text { Aug } & .07 \\\\\hline \text { Sep } & .2 \\\\\hline \text { Oct } & 1.04 \\\\\hline \text { Nov } & 2.49 \\\\\hline \text { Dec } & 2.89 \\\\\hline\end{array}$$

On the basis of the results of Exercises \(37-42,\) under what conditions on the constants \(a, k, h, d, r, s\) does it appear that the graph of $$f(t)=a \sin (k t+h)+d \cos (r t+s)$$ coincides with the graph of the function $$g(t)=A \sin (b t+c) ?$$

Explore various ways in which a calculator can produce inaccurate graphs of trigonometric functions. These exercises also provide examples of two functions, with different graphs, whose graphs appear identical in certain viewing windows. Find a viewing window in which the graphs of \(y=\cos x\) and \(y=.54\) appear identical. [Hint: See the chart in Exercise 61 and note that \(\cos 1 \approx .54 .]\)

Show that the given function is periodic with period less than \(2 \pi\). [Hint: Find a positive number \(k\) with \(k<2 \pi \text { such that } f(t+k)=f(t) \text { for every t in the domain of } f .]\) $$f(t)=\sin 4 t$$

In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\sec t+\csc t}{1+\tan t}=\csc t$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). \(t\) minutes, \(k\) rpm

(a) Find two numbers \(c\) and \(d\) such that $$ \sin (c+d) \neq \sin c+\sin d $$ (b) Find two numbers \(c\) and \(d\) such that $$ \cos (c+d) \neq \cos c+\cos d $$ (TABLE CAN'T COPY)

A grandfather clock has a pendulum length of \(k\) meters and its swing is given (as in Exercise 57 ) by the function \(f(t)=.25 \sin (\omega t),\) where $$\omega=\sqrt{\frac{9.8}{k}}$$ (a) Find \(k\) such that the period of the pendulum is 2 seconds. (b) The temperature in the summer months causes the pendulum to increase its length by \(.01 \% .\) How much time will the clock lose in June, July, and August? [Hint: These three months have a total of 92 days (7,948,800 seconds). If \(k \text { is increased by } .01 \%, \text { what is } f(2) ?]\)

Provide further examples of functions with different graphs, whose graphs appear identical in certain viewing windows. Approximating trigonometric functions by polynomials. For each odd positive integer \(n,\) let \(f_{n}\) be the function whose rule is $$ f_{n}(t)=t-\frac{t^{3}}{3 !}+\frac{t^{5}}{5 !}-\frac{t^{7}}{7 !}+\cdots-\frac{t^{n}}{n !} $$ since the signs alternate, the sign of the last term might be \+ instead of \(-,\) depending on what \(n\) is. Recall that \(n !\) is the product of all integers from 1 to \(n\); for instance, \(5 !=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5=120\) (a) Graph \(f_{7}(t)\) and \(g(t)=\sin t\) on the same screen in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) For what values of \(t\) does \(f_{7}\) appear to be a good approximation of \(g ?\) (b) What is the smallest value of \(n\) for which the graphs of \(f_{n}\) and \(g\) appear to coincide in this window? In this case, determine how accurate the approximation is by finding \(f_{n}(2)\) and \(g(2)\)

Show that the given function is periodic with period less than \(2 \pi\). [Hint: Find a positive number \(k\) with \(k<2 \pi \text { such that } f(t+k)=f(t) \text { for every t in the domain of } f .]\) $$f(t)=\sin (\pi t)$$

In Exercises \(65-70,\) draw a rough sketch to determine if the given number is positive. sin \(1[\text { Hint: The terminal side of an angle of } 1\) radian lies in the first quadrant (why?), so any point on it will have a positive y-coordinate.

Show that the given function is periodic with period less than \(2 \pi\). [Hint: Find a positive number \(k\) with \(k<2 \pi \text { such that } f(t+k)=f(t) \text { for every t in the domain of } f .]\) $$f(t)=\cos (3 \pi t / 2)$$

In Exercises \(65-70,\) draw a rough sketch to determine if the given number is positive. $$\cos 2$$

Provide further examples of functions with different graphs, whose graphs appear identical in certain viewing windows. Find a rational function whose graph appears to coincide with the graph of \(h(t)=\tan t\) when $$ -2 \pi \leq t \leq 2 \pi $$ \([\text {Hint: Exercises } 65 \text { and } 66 .]\)

Show that the given function is periodic with period less than \(2 \pi\). [Hint: Find a positive number \(k\) with \(k<2 \pi \text { such that } f(t+k)=f(t) \text { for every t in the domain of } f .]\) $$f(t)=\tan 2 t$$

Provide further examples of functions with different graphs, whose graphs appear identical in certain viewing windows. Find a periodic function whose graph consists of "square waves." [Hint: Consider the sum $$ \left.\sin \pi t+\frac{1}{3} \sin 3 \pi t+\frac{1}{5} \sin 5 \pi t+\frac{1}{7} \sin 7 \pi t+\cdots\right] $$

Fill the blanks with "even" or "odd" so that the resulting statement is true. Then prove the statement by using an appropriate identity. [Hint: Special Topics \(3.4 .\) A may be helpful.] (a) \(f(t)=\sin t\) is an _____ function. (b) \(g(t)=\cos t\) is an _____ function. (c) \(h(t)=\tan t\) is an _____ function. (d) \(f(t)=t \sin t\) is an _____ function. (e) \(g(t)=t+\tan t\) is an _____ function.

In Exercises \(65-70,\) draw a rough sketch to determine if the given number is positive. $$(\cos 2)(\sin 2)$$

With your calculator in parametric graphing mode and the range values $$ 0 \leq t \leq 6.28 \quad-1 \leq x \leq 6.28 \quad-2.5 \leq y \leq 2.5 $$ graph the following two functions on the same screen: $$ x_{1}=\cos t, y_{1}=\sin t \quad \text { and } \quad x_{2}=t, y_{2}=\cos t $$ Using the trace feature, move the cursor along the first graph (the unit circle). Stop at a point on the circle, note the value of \(t\) and the \(x\) -coordinate of the point. Then switch the trace to the second graph (the cosine function) by using the up or down cursor arrows. The value of \(t\) remains the same. How does the \(y\) -coordinate of the new point compare with the \(x\) -coordinate of the original point on the unit circle? Explain what's going on.