Chapter 5

Finding the Domain of a Function Find the domain of the function. $$f(x)=-x^{2}-1$$

Identify the rule of algebra illustrated by the statement. \(7\left(\frac{1}{7}\right)=1\)

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) for all real numbers \(x\) in the interval \([-2 \pi, 2 \pi]\) Use the graphs to find the real numbers \(x\) such that \(y_{1}=y_{2}.\) $$\begin{aligned} &y_{1}=\cos x\\\ &y_{2}=\frac{1}{2} \end{aligned}$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\cos \left(\arcsin \frac{x-h}{r}\right)\)

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$10 \pi / 3$$

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-4.2 \pi$$

Finding the Domain of a Function Find the domain of the function. $$g(x)=\sqrt[3]{x+2}$$

In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is \(3.5^{\circ} .\) After you drive 13 miles closer to the mountain, the angle of elevation is \(9^{\circ}\) (see figure). Approximate the height of the mountain.

Identify the rule of algebra illustrated by the statement. \((3+x)+0=3+x\)

Complete the equation $$\arctan \frac{14}{x}=\arcsin (\square), \quad x>0$$

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-17 \pi / 6$$

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-2$$

Finding the Domain of a Function Find the domain of the function. $$g(x)=\sqrt{7-x}$$

Identify the rule of algebra illustrated by the statement. \((a+b)+10=a+(b+10)\)

For a person at rest, the velocity \(v\) (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by $$v=0.85 \sin \frac{\pi t}{3}$$ where \(t\) is the time (in seconds). (Inhalation occurs when \(v>0,\) and exhalation occurs when \(v<0 .\) ) (a) Use a graphing utility to graph \(v\). (b) Find the time for one full respiratory cycle. (c) Find the number of cycles per minute. (d) The model is for a person at rest. How might the model change for a person who is exercising? Explain.

Complete the equation $$\arcsin \frac{\sqrt{36-x^{2}}}{6}=\arccos (\square), \quad 0 \leq x \leq 6$$

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-20 \pi / 3$$

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-0.57$$

A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is 16 feet from the streetlight and 5 feet from the tip of the streetlight's shadow, the person's shadow starts to appear beyond the streetlight's shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the streetlight. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the streetlight?

Determine whether the function is one-to-one. If it is, find its inverse function. \(f(x)=-10\)

A company that produces snowboards, which are seasonal products, forecasts monthly sales for one year to be $$S=74.50+43.75 \cos \frac{\pi t}{6}$$ where \(S\) is the sales in thousands of units and \(t\) is the time in months, with \(t=1\) corresponding to January. (a) Use a graphing utility to graph the sales function over the one-year period. (b) Use the graph in part (a) to determine the months of maximum and minimum sales.

Complete the equation \(\arccos \frac{3}{\sqrt{x^{2}-2 x+10}}=\arcsin ( \square\) )

Finding the Domain of a Function Find the domain of the function. $$h(x)=\frac{x}{x^{2}-9}$$

A 20 -meter line is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of approximately \(85^{\circ}\) with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the balloon? (d) The breeze becomes stronger, and the angle the balloon makes with the ground decreases. How does this affect your triangle from part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures \(\theta\). $$\begin{array}{|l|l|l|l|l|}\hline \text { Angle, } \theta & 80^{\circ} & 70^{\circ} & 60^{\circ} & 50^{\circ} \\\\\hline \text { Height } & & & & \\\\\hline\end{array}$$ $$\begin{array}{|l|l|l|l|l|}\hline \text { Angle, } \theta & 40^{\circ} & 30^{\circ} & 20^{\circ} & 10^{\circ} \\\\\hline \text { Height } & & & & \\\\\hline\end{array}$$ (f) As the angle the balloon makes with the ground approaches \(0^{\circ},\) how does this affect the height of the balloon? Draw a right triangle to explain your reasoning.

The daily consumption \(C\) (in gallons) of diesel fuel on a farm is modeled by $$C=30.3+21.6 \sin \left(\frac{2 \pi t}{365}+10.9\right)$$ where \(t\) is the time in days, with \(t=1\) corresponding to January 1. (a) What is the period of the model? Is it what you expected? Explain. (b) What is the average daily fuel consumption? Which term of the model did you use? Explain. (c) Use a graphing utility to graph the model. Use the graph to approximate the time of the year when consumption exceeds 40 gallons per day.

Determine whether the statement is true or false. Justify your answer. $$\sin 60^{\circ} \csc 60^{\circ}=1$$

Determine whether the function is one-to-one. If it is, find its inverse function. \(f(x)=\sqrt{3 x-14}\)

You are riding a Ferris wheel. Your height \(h\) (in feet) above the ground at any time \(t\) (in seconds) can be modeled by $$h=25 \sin \frac{\pi}{15}(t-75)+30$$ The Ferris wheel turns for 135 seconds before it stops to let the first passengers off. (a) Use a graphing utility to graph the model. (b) What are the minimum and maximum heights above the ground?

Use a graphing utility to graph the function. \(y=2 \arccos x\)

Determine two coterminal angles in radian measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(\frac{9 \pi}{4}\) (b) \(-\frac{2 \pi}{15}\)

Determine whether the statement is true or false. Justify your answer. $$\sec 30^{\circ}=\csc 60^{\circ}$$

Determine whether the function is one-to-one. If it is, find its inverse function. \(f(x)=\sqrt[4]{x-5}\)

Use a graphing utility to graph the function. \(y=\arcsin \frac{x}{2}\)

Determine two coterminal angles in radian measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(-\frac{7 \pi}{8}\) (b) \(\frac{\pi}{12}\)

Determine whether the statement is true or false. Justify your answer. $$\sin 45^{\circ}+\cos 45^{\circ}=1$$

The table shows the percent \(y\) (in decimal form) of the moon's face that is illuminated on day \(x\) of the year \(2016,\) where \(x=1\) represents January 1. $$\begin{array}{|c|c|} \hline \text { Day,\(x\) } & \text { Percent,\(y\) } \\\ \hline 10 & 0.0 \\ 16 & 0.5 \\ 24 & 1.0 \\ 32 & 0.5 \\ 39 & 0.0 \\ 46 & 0.5 \end{array}$$ (a) Create a scatter plot of the data. (b) Find a trigonometric model for the data. (c) Add the graph of your model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the percent illumination of the moon on June 21,2017 . (Assume there are 366 days in 2016 .)

Identify the domain, any intercepts, and any asymptotes of the function. \(y=x^{2}+3 x-4\)

Use a graphing utility to graph the function. \(f(x)=\arcsin (x-2)\)

Find (if possible) the complement and supplement of the angle. $$\frac{\pi}{3}$$

The table shows the average daily high temperatures (in degrees Fahrenheit) for Quillayute, Washington, \(Q\) and Chicago, Illinois, \(C\) for month \(t,\) with \(t=1\) corresponding to January. $$\begin{array}{c|c|c} \text { Month, } & \text { Quillayute, } & \text { Chicago, } \\ t & Q & C \\ \hline 1 & 47.1 & 31.0 \\ 2 & 49.1 & 35.3 \\ 3 & 51.4 & 46.6 \\ 4 & 54.8 & 59.0 \\ 5 & 59.5 & 70.0 \\ 6 & 63.1 & 79.7 \\ 7 & 67.4 & 84.1 \\ 8 & 68.6 & 81.9 \\ 9 & 66.2 & 74.8 \\ 10 & 58.2 & 62.3 \\ 11 & 50.3 & 48.2 \\ 12 & 46.0 & 34.8 \end{array}$$ (a) \(\mathrm{A}\) model for the temperature in Quillayute is given by $$Q(t)=57.5+10.6 \sin (0.566 x-2.568)$$ Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data and the model for the temperatures in Quillayute in the same viewing window. How well does the model fit the data? (c) Use the graphing utility to graph the data and the model for the temperatures in Chicago in the same viewing window. How well does the model fit the data? (d) Use the models to estimate the average daily high temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.

Determine whether the statement is true or false. Justify your answer. $$\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ}=-1$$

Identify the domain, any intercepts, and any asymptotes of the function. \(y=\ln x^{4}\)

Use a graphing utility to graph the function. \(g(t)=\arccos (t+2)\)

Find (if possible) the complement and supplement of the angle. $$\frac{3 \pi}{4}$$

Determine whether the statement is true or false. Justify your answer. The graph of the function given by \(g(x)=\sin (x+2 \pi)\) translates the graph of \(f(x)=\sin x\) one period to the right.

Identify the domain, any intercepts, and any asymptotes of the function. \(f(x)=3^{x+1}+2\)

Use a graphing utility to graph the function. \(f(x)=\arctan \frac{x}{4}\)

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\sin \theta=\frac{2}{5}, \cos \theta > 0$$

Find (if possible) the complement and supplement of the angle. $$\frac{2 \pi}{3}$$

Determine whether the statement is true or false. Justify your answer. The graph of \(y=6-\frac{3}{4} \sin \frac{3 x}{10}\) has a period of \(\frac{20 \pi}{3}.\)