Chapter 9

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\tan s<0, \sin s>0$$

The maximum monthly average temperature in Anchorage, Alaska, is \(57^{\circ} \mathrm{F}\) and the minimum is \(12^{\circ} \mathrm{F}\). $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Month } & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline \text { Temperature ('F) } & 12 & 18 & 23 & 36 & 46 & 55\end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Month } & 7 & 8 & 9 & 10 & 11 & 12 \\\\\hline \text { Temperature ('F) } & 57 & 55 & 48 & 36 & 23 & 16\end{array}$$ (a) Using only these two temperatures, determine \(f(x)=a \cos [b(x-c)]+d\) so that \(f(x)\) models the monthly average temperatures in Anchorage. (b) Graph \(f\) and the actual data in the table over a 2 -year period.

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\sin \frac{5 \pi}{3}$$

$$\text { Prove that if } \cos s \neq 0, \text { then } 1+\tan ^{2} s=\sec ^{2} s$$

Find the length of each arc intercepted by a central angle \(\boldsymbol{\theta}\) in a circle of radius \(\kappa\) Round answers to the nearest hundredth. $$r=12.3 \text { centimeters; } \theta=\frac{2 \pi}{3} \text { radians }$$

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\cos \frac{7 \pi}{6}$$

Explain why there is no angle \(\theta\) that satisfies \(\tan \theta>0, \cot \theta<0\)

$$\text { Prove that if } \sin s \neq 0, \text { then } 1+\cot ^{2} s=\csc ^{2} s$$.

Find the length of each arc intercepted by a central angle \(\boldsymbol{\theta}\) in a circle of radius \(\kappa\) Round answers to the nearest hundredth. $$r=0.892 \text { centimeter; } \theta=\frac{11 \pi}{10} \text { radians }$$

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\tan \frac{4 \pi}{3}$$

Decide whether each statement is possible for some angle \(\boldsymbol{\theta}\), or impossible for that angle. $$\sin \theta=2$$

The tables give the fractional part of the moon that is illuminated during the month indicated. (a) Plot the data for the month. (b) Use sine regression to determine a model for the data. (c) Graph the equation from part (b) together with the data on the same coondinate axes. January 2015. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\\\\hline \text { Fraction } & 0.84 & 0.91 & 0.96 & 0.99 & 1.00 & 0.99 & 0.96 & 0.92 & 0.86 & 0.79 & 0.70 & 0.62 & 0.52 & 0.42 & 0.33 & 0.23\end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Day } & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 \\\\\hline \text { Fraction } & 0.15 & 0.08 & 0.03 & 0.00 & 0.01 & 0.04 & 0.10 & 0.19 & 0.28 & 0.39 & 0.50 & 0.61 & 0.71 & 0.80 & 0.87\end{array}$$

Find the length of each arc intercepted by a central angle \(\boldsymbol{\theta}\) in a circle of radius \(\kappa\) Round answers to the nearest hundredth. $$r=4.82 \text { meters; } \theta=60^{\circ}$$

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\sin \theta=\frac{1}{2}$$

The tables give the fractional part of the moon that is illuminated during the month indicated. (a) Plot the data for the month. (b) Use sine regression to determine a model for the data. (c) Graph the equation from part (b) together with the data on the same coondinate axes. November 2015 $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\\\\hline \text { Fraction } & 0.73 & 0.63 & 0.53 & 0.43 & 0.34 & 0.25 & 0.18 & 0.11 & 0.06 & 0.02 & 0.00 & 0.00 & 0.02 & 0.06\end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Day } & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\\\\hline \text { Fraction } & 0.12 & 0.19 & 0.28 & 0.39 & 0.49 & 0.61 & 0.71 & 0.81 & 0.90 & 0.96 & 0.99 & 1.00 & 0.98 & 0.93 & 0.87 & 0.79\end{array}$$

Find the length of each arc intercepted by a central angle \(\boldsymbol{\theta}\) in a circle of radius \(\kappa\) Round answers to the nearest hundredth. $$r=71.9 \text { centimeters; } \theta=135^{\circ}$$

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\sin \theta=\frac{\sqrt{2}}{2}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cos \theta=-0.96$$

Let \(s\) be a real number corresponding to the point ( \(a, b\) ) on the unit circle. Use this information to determine an expression representing the sine and cosine of each real number. $$s-6 \pi$$

Tides cause ocean currents to flow into and out of harbors and canals. The table shows the speed of the ocean current at Cape Cod Canal in bogo-knots (bk) \(x\) hours after midnight on August 26 \(1998 .\) (To change bogo-knots to knots, take the square root of the absolute value of the number of bogo- knots.) $$\begin{array}{|l|c|c|c|c|c|c}\text { Time (hr) } & 3.7 & 6.75 & 9.8 & 13.0 & 16.1 & 22.2 \\\\\hline \text { Current (bk) } & -18 & 0 & 18 & 0 & -18 & 18\end{array}$$ (a) Use \(f(x)=a \cos [b(x-c)]+d\) to model the data. (b) Graph \(f\) and the data in \([0,24]\) by \([-20,20]\) Interpret the graph.

Find the distance in kilometers. between the pair of cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of Earth is 6400 kilometers. Round answers to the nearest hundred kilometers. Madison, South Dakota, \(44^{\circ} \mathrm{N},\) and Dallas, Texas, \(33^{\circ} \mathrm{N}\)

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\tan \theta=\sqrt{3}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cos \theta=-0.56$$

Let \(s\) be a real number corresponding to the point ( \(a, b\) ) on the unit circle. Use this information to determine an expression representing the sine and cosine of each real number. $$s+\pi$$

Find the distance in kilometers. between the pair of cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of Earth is 6400 kilometers. Round answers to the nearest hundred kilometers. Charleston, South Carolina, \(33^{\circ} \mathrm{N},\) and Toronto, Ontario, \(43^{\circ} \mathrm{N}\)

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\tan \theta=\frac{\sqrt{3}}{3}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\tan \theta=0.93$$

Let \(s\) be a real number corresponding to the point ( \(a, b\) ) on the unit circle. Use this information to determine an expression representing the sine and cosine of each real number. $$-s$$

Atmospheric Carbon Dioxide The carbon dioxide content in the atmosphere at Barrow, Alaska, in parts per million (ppm) can be modeled with the function $$C(x)=0.04 x^{2}+0.6 x+330+7.5 \sin (2 \pi x)$$ where \(x\) is in years and where \(x=0\) corresponds to \(1970 .\) (Source: Zeilik, M., S. Gregory, and E. Smith. Introductory Astronomy and Astrophysics, Fourth Edition, Saunders College Publishers.) (a) Graph \(C\) for \(5 \leq x \leq 25\). (Hint: For the range, use \(320 \leq y \leq 380\) (b) Define a new function \(C\) that is valid if \(x\) represents the actual year, where \(1970 \leq x \leq 1995\)

Find the distance in kilometers. between the pair of cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of Earth is 6400 kilometers. Round answers to the nearest hundred kilometers. New York City, New York, 41 \(^{\circ} \mathrm{N}\), and Lima, Peru, \(12^{\circ} \mathrm{S}\)

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\cos \theta=\frac{\sqrt{3}}{2}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cot \theta=0.93$$

At Mauna Loa, Hawaii, atmospheric carbon dioxide levels in parts per million (ppm) have been measured regularly since 1958 . The function $$L(x)=0.022 x^{2}+0.55 x+316+3.5 \sin (2 \pi x)$$ can be used to model these levels, where \(x\) is in years and \(x=0\) corresponds to \(1960 .\) (Source: Nilsson, A., Greenhouse Earth. John Wiley and Sons.) (a) Graph \(L\) for \(15 \leq x \leq 35 .\) (Hint: For the range, use \(325 \leq y \leq 365\) (b) When do the seasonal maximum and minimum carbon dioxide levels occur? (c) \(L\) is the sum of a quadratic function and a sine function. What is the significance of each of these functions? Discuss what physical phenomena may be responsible for each function.

Find the distance in kilometers. between the pair of cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of Earth is 6400 kilometers. Round answers to the nearest hundred kilometers. Halifax, Nova Scotia, \(45^{\circ} \mathrm{N},\) and Buenos Aires, Argentina, \(34^{\circ} \mathrm{S}\)

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\sin \theta=\frac{\sqrt{2}}{2}$$

Use the formula \(\omega=\frac{\theta}{t}\) to find the value of the missing variable. Round to the nearest thousandth. $$\theta=\frac{3 \pi}{4} \text { radians, } t=8 \text { seconds }$$

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\sin \theta=\frac{1}{2}$$

Use the formula \(\omega=\frac{\theta}{t}\) to find the value of the missing variable. Round to the nearest thousandth. $$\theta=\frac{2 \pi}{s} \text { radians, } t=10 \text { seconds }$$

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\cos \theta=\frac{\sqrt{3}}{2}$$

(a) Compare the graphs of $$y=\sin 2 x \text { and } y=2 \sin x$$ over the interval \([0,2 \pi] .\) Can we say that, in general, \(\sin b x=b \sin x ?\) Explain. (b) Compare the graphs of $$y=\cos 3 x \text { and } y=3 \cos x$$ over the interval \([0,2 \pi] .\) Can we say that, in general, \(\cos b x=b \cos x ?\) Explain.

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\csc \theta=100$$

Use the formula \(\omega=\frac{\theta}{t}\) to find the value of the missing variable. Round to the nearest thousandth. \(\theta=\frac{2 \pi}{9}\) radian, \(\omega=\frac{5 \pi}{27}\) radian per minute

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\tan \theta=-\sqrt{3}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\csc \theta=-100$$

Use the formula \(\omega=\frac{\theta}{t}\) to find the value of the missing variable. Round to the nearest thousandth. \(\omega=0.90674\) radian per minute, \(t=11.876\) minutes

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\sec \theta=-\sqrt{2}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cot \theta=-4$$

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\) wt. Substituting wt for \(\theta\) changes \(s=r \theta\) to \(s=r \omega t\). Use the formula \(s=r \omega t\) to find the value of the missing variable. \(r=6\) centimeters, \(\omega=\frac{\pi}{3}\) radians per second, \(t=9\) seconds

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\cot \theta=-\frac{\sqrt{3}}{3}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cot \theta=-6$$