Chapter 5

Identify the domain, any intercepts, and any asymptotes of the function. \(f(x)=\frac{x-7}{x^{2}+4 x+4}\)

Use a graphing utility to graph the function. \(f(x)=\arctan 3 x\)

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\cos \theta=-\frac{3}{7}, \sin \theta < 0$$

Find (if possible) the complement and supplement of the angle. $$\frac{\pi}{6}$$

You are given the value of tan \(\theta .\) Is it possible to find the value of \(\sec \theta\) without finding the measure of \(\theta ?\) Explain.

Write the function in terms of the sine function by using the identity \(A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right)\) Use a graphing utility to graph both forms of the function. What does the graph imply? \(f(t)=3 \cos 2 t+3 \sin 2 t\)

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\tan \theta=-4, \cos \theta < 0$$

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

Determine whether the statement is true or false. Justify your answer. The graph of \(y=-\cos x\) is a reflection of the graph of \(y=\sin \left(x+\frac{\pi}{2}\right)\) in the \(x\) -axis.

Write the function in terms of the sine function by using the identity \(A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right)\) Use a graphing utility to graph both forms of the function. What does the graph imply? \(f(t)=4 \cos \pi t+3 \sin \pi t\)

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\cot \theta=5, \sin \theta > 0$$

Find (if possible) the complement and supplement of the angle. $$\frac{12 \pi}{5}$$

Sketch the graph of \(y=\cos b x\) for \(b=\frac{1}{2}, 2\) and \(3 .\) How does the value of \(b\) affect the graph? How many complete cycles of the graph of \(y\) occur between 0 and \(2 \pi\) for each value of \(b ?\)

(a) Use a graphing utility to complete the table. Round your results to four decimal places. $$\begin{array}{|l|l|l|l|l|l|}\hline \theta & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} \\\\\hline \sin \theta & & & & & \\\\\hline \cos \theta & & & & & \\\\\hline \tan \theta & & & & & \\\\\hline\end{array}$$ (b) Classify each of the three trigonometric functions as increasing or decreasing for the table values. (c) From the values in the table, verify that the tangent function is the quotient of the sine and cosine functions.

Find the value. If not possible, state the reason. 93\. As \(x \rightarrow 1^{-},\) the value of arcsin \(x \rightarrow\)

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\csc \theta=-\frac{3}{2}, \tan \theta < 0$$

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius \(r\) 15 inches Arc Length \(s\) 8 inches

Use a graphing utility to graph the function given by \(y=d+a \sin (b x-c)\) for several different values of \(a, b, c,\) and \(d .\) Write a paragraph describing how the values of \(a, b, c,\) and \(d\) affect the graph.

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=e^{3 x}$$

Find the value. If not possible, state the reason. As \(x \rightarrow 1^{-},\) the value of arccos \(x \rightarrow\)

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\sec \theta=-\frac{4}{3}, \cot \theta > 0$$

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius \(r\) 22 feet Arc Length \(s\) 10 feet

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=-e^{3 x}$$

Find the value. If not possible, state the reason. As \(x \rightarrow \infty,\) the value of arctan \(x \rightarrow\)

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\sin 105^{\circ}$$

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius \(r\) 14.5 centimeters Arc Length \(s\) 35 centimeters

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=2+e^{3 x}$$

Find the value. If not possible, state the reason. As \(x \rightarrow-1^{+},\) the value of arcsin \(x \rightarrow\)

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\sec 235^{\circ}$$

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius \(r\) 80 kilometers Arc Length \(s\) 160 kilometers

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=-4+e^{3 x}$$

Find the value. If not possible, state the reason. As \(x \rightarrow-1^{+},\) the value of arccos \(x \rightarrow\)

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\cos \left(-110^{\circ}\right)$$

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 14 inches Central Angle \(\theta\) \(\pi\) radians

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=\log _{3} x$$

Find the value. If not possible, state the reason. As \(x \rightarrow-\infty,\) the value of arctan \(x \rightarrow\)

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\sin \left(-220^{\circ}\right)$$

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 9 feet Central Angle \(\theta\) \(\frac{\pi}{3}\) radians

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=\log _{3} x+1$$

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\tan (2 \pi / 9)$$

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 27 meters Central Angle \(\theta\) \(120^{\circ}\)

Use a graphing utility to explore the ratio \((1-\cos x) / x,\) which appears in calculus. (a) Complete the table. Round your results to four decimal places. (b) Use the graphing utility to graph the function \(f(x)=\frac{1-\cos x}{x}\). Use the zoom and trace features to describe the behavior of the graph as \(x\) approaches \(0 .\) (c) Write a brief statement regarding the value of the ratio based on your results in parts (a) and (b).

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=-\log _{3} x$$

A photographer takes a picture of a three-foot painting hanging in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle \(\beta\) subtended by the camera lens \(x\) feet from the painting is \(\beta=\arctan \left[3 x /\left(x^{2}+4\right)\right], x>0\) (a) Use a graphing utility to graph \(\beta\) as a function of \(x .\) (b) Use the trace feature to approximate the distance from the picture when \(\beta\) is maximum. (c) Identify the asymptote of the graph and discuss its meaning in the context of the problem.

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\tan (11 \pi / 9)$$

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 12 centimeters Central Angle \(\theta\) \(135^{\circ}\)

Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials $$\sin x \approx x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \quad \text { and } \quad \cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}$$ where \(x\) is in radians. (a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare? (c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added?

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=\log _{3}(x-4)$$

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\csc (-8 \pi / 9)$$

Find the radius \(r\) of a circle with an arc length \(s\) and a central angle \(\theta\). Arc Length \(s\) 36 feet Central Angle \(\theta\) \(\frac{\pi}{2}\) radians