Chapter 5

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \sin & 30^{\circ} & \square & \square \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{array}{l} f(x)=\sin x \\ g(x)=\sin (x-\pi) \end{array}$$

Sketch each angle in standard position. (a) \(-30^{\circ}\) (b) \(150^{\circ}\)

The front of an A-frame cottage has the shape of an isosceles triangle. It stands 28 feet high and the angle of elevation of its roof is \(70^{\circ} .\) What is the width of the cottage at its base?

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arctan 15

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\frac{1}{2} \tan \pi x\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-24,10)$$

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \cos & 60^{\circ} & \square & \square \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{array}{l} f(x)=\cos x \\ g(x)=\cos (x+\pi) \end{array}$$

Sketch each angle in standard position. (a) \(270^{\circ}\) (b) \(-120^{\circ}\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=-\csc (\pi-x)\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-\sqrt{21}, 2)$$

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \tan & \square & \frac{\pi}{3} & \square \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\cos 2 x\\\ &g(x)=-\cos 2 x \end{aligned}$$

Sketch each angle in standard position. (a) \(405^{\circ}\) (b) \(-780^{\circ}\)

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\sin ^{-1} 0.56\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=-\sec (x+\pi)\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-3,-\sqrt{7})$$

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \sec & \square & \frac{\pi}{4} & \square \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\sin 3 x\\\ &g(x)=\sin (-3 x) \end{aligned}$$

Sketch each angle in standard position. (a) \(-450^{\circ}\) (b) \(600^{\circ}\)

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arccos( -0.9 )

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\frac{1}{2} \sec (2 x-\pi)\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-3,-\sqrt{7})$$

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \csc & \square & \square & \sqrt{2} \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{array}{l} f(x)=\cos 2 x \\ g(x)=3+\cos 2 x \end{array}$$

Architecture From a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are \(37^{\circ}\) and \(49^{\circ} 40^{\prime},\) respectively. (a) Draw right triangles that give a visual representation of the problem. Label the known and unknown quantities. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) Find the height of the steeple.

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\csc (2 x-\pi)\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(3,-9)$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\cos 4 x\\\ &g(x)=-2+\cos 4 x \end{aligned}$$

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arctan (-6)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=2 \cot \left(x+\frac{\pi}{2}\right)\)

State the quadrant in which \(\theta\) lies. $$\sin \theta < 0 \text { and } \cos \theta < 0$$

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \cos & \square & \frac{\pi}{6} & \square \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{array}{l} f(x)=\sin x \\ g(x)=5 \sin (-x) \end{array}$$

Determine two coterminal angles in degree measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(300^{\circ}\) (b) \(-740^{\circ}\)

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\tan ^{-1} 5.9\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\frac{1}{4} \cot (x+\pi)\)

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \sin & \square & \frac{\pi}{4} & \square \end{array}$$

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=-\frac{1}{2} \sin x \end{aligned}$$

Determine two coterminal angles in degree measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(-445^{\circ}\) (b) \(230^{\circ}\)

The angle of elevation to a plane approaching your home is \(16^{\circ}\). One minute later, it is \(57^{\circ}\). You assume that the speed of the plane is 550 miles per hour. Approximate the altitude of the plane.

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\sin ^{-1}\left(-\frac{3}{4}\right)\)

Use a graphing utility to graph the function (include two full periods). Graph the corresponding reciprocal function in the same viewing window. Describe and compare the graphs. \(y=2 \csc 3 x\)

State the quadrant in which \(\theta\) lies. $$\cot \theta > 0 \text { and } \cos \theta > 0$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$64^{\circ} 45^{\prime}$$

While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the peak is \(2.5^{\circ} .\) After you drive 18 miles closer to the mountain, the angle of elevation is \(10^{\circ}\). Approximate the height of the mountain.

Use a graphing utility to graph the function (include two full periods). Graph the corresponding reciprocal function in the same viewing window. Describe and compare the graphs. \(y=-\csc 4 x\)

Construct an appropriate triangle to complete the table. \(\left(0^{\circ} \leq \theta \leq 90^{\circ}, 0 \leq \theta \leq \pi / 2\right)\) $$\begin{array}{llll} \text {Function} & \theta(\operatorname{deg}) & \theta(\mathrm{rad}) & \text {Function Value} \\ \tan & \square & \square & \frac{1}{\sqrt{3}} \end{array}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$-124^{\circ} 30^{\prime}$$