Data Analysis: Meteorology The times of sunset (Greenwich Mean Time) at 40 north

In Exercises 1 and 2,estimate the angle to the nearest one-half radian.

In Exercises 1 and 2,estimate the angle to the nearest one-half radian.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 310, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle.

In Exercises 1114, convert the angle measure from degrees to radians. Round your answer to three decimal places.

In Exercises 1114, convert the angle measure from degrees to radians. Round your answer to three decimal places.

In Exercises 1114, convert the angle measure from degrees to radians. Round your answer to three decimal places.

In Exercises 1114, convert the angle measure from degrees to radians. Round your answer to three decimal places.

In Exercises 1518, convert the angle measure from radians to degrees. Round your answer to three decimal places.

In Exercises 1518, convert the angle measure from radians to degrees. Round your answer to three decimal places.

In Exercises 1518, convert the angle measure from radians to degrees. Round your answer to three decimal places.

In Exercises 1518, convert the angle measure from radians to degrees. Round your answer to three decimal places.

Arc Length Find the length of the arc on a circle with a radius of 20 inches intercepted by a central angle of 2

Arc Length Find the length of the arc on a circle with a radius of 11 meters intercepted by a central angle of 2

Phonograph Compact discs have all but replaced phonograph records. Phonograph records are vinyl discs that rotate on a turntable. A typical record album is 12 inches in diameter and plays at 33 revolutions per minute. (a) What is the angular speed of a record album? (b) What is the linear speed of the outer edge of a record album? 2

Bicycle At what speed is a bicyclist traveling when his 27-inch-diameter tires are rotating at an angular speed of radians per second? 2

Circular Sector Find the area of the sector of a circle with a radius of 18 inches and central angle 2

Circular Sector Find the area of the sector of a circle with a radius of 6.5 millimeters and central angle 2

In Exercises 2528, find the point on the unit circle that corresponds to the real numbe

In Exercises 2528, find the point on the unit circle that corresponds to the real numbe

In Exercises 2528, find the point on the unit circle that corresponds to the real numbe

In Exercises 2528, find the point on the unit circle that corresponds to the real numbe

In Exercises 2932, evaluate (if possible) the six trigonometric functions of the real number.

In Exercises 2932, evaluate (if possible) the six trigonometric functions of the real number.

In Exercises 2932, evaluate (if possible) the six trigonometric functions of the real number.

In Exercises 2932, evaluate (if possible) the six trigonometric functions of the real number.

In Exercises 3336, evaluate the trigonometric function using its period as an aid.

In Exercises 3336, evaluate the trigonometric function using its period as an aid.

In Exercises 3336, evaluate the trigonometric function using its period as an aid.

In Exercises 3336, evaluate the trigonometric function using its period as an aid.

In Exercises 3740, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 3740, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 3740, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 3740, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 4144, find the exact values of the six trigonometric functions of the angle shown in the figure.

In Exercises 4144, find the exact values of the six trigonometric functions of the angle shown in the figure.

In Exercises 4144, find the exact values of the six trigonometric functions of the angle shown in the figure.

In Exercises 4144, find the exact values of the six trigonometric functions of the angle shown in the figure.

In Exercises 45 48, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions.

In Exercises 45 48, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions.

In Exercises 45 48, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions.

In Exercises 45 48, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions.

In Exercises 4954, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 4954, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 4954, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 4954, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 4954, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 4954, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

Railroad Grade A train travels 3.5 kilometers on a straight track with a grade of (see figure). What is the vertical rise of the train in that distance? 5

Guy Wire A guy wire runs from the ground to the top of a 25-foot telephone pole. The angle formed between the wire and the ground is How far from the base of the pole is the wire attached to the ground? 5

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 5764, 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

In Exercises 6570, find the values of the six trigonometric functions of

In Exercises 6570, find the values of the six trigonometric functions of

In Exercises 6570, find the values of the six trigonometric functions of

In Exercises 6570, find the values of the six trigonometric functions of

In Exercises 6570, find the values of the six trigonometric functions of

In Exercises 6570, find the values of the six trigonometric functions of

In Exercises 7174, find the reference angle and sketch and in standard position.

In Exercises 7174, find the reference angle and sketch and in standard position.

In Exercises 7174, find the reference angle and sketch and in standard position.

In Exercises 7174, find the reference angle and sketch and in standard position.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 7582, evaluate the sine, cosine, and tangent of the angle without using a calculator.

In Exercises 8388, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 8388, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 8388, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 8388, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 8388, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 8388, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

In Exercises 8996, sketch the graph of the function. Include two full periods.

Sound Waves Sound waves can be modeled by sine functions of the form where is measured in seconds. (a) Write an equation of a sound wave whose amplitude is 2 and whose period is second. (b) What is the frequency of the sound wave described in part (a)? 9

Data Analysis: Meteorology The times of sunset (Greenwich Mean Time) at 40 north latitude on the 15th of each month are: 1(16:59), 2(17:35), 3(18:06), 4(18:38), 5(19:08), 6(19:30), 7(19:28), 8(18:57), 9(18:09), 10(17:21), 11(16:44), 12(16:36). The month is represented by with corresponding to January. A model (in which minutes have been converted to the decimal parts of an hour) for the data is (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the model? Explain. 9

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 99106, sketch a graph of the function. Include two full periods.

In Exercises 107 and 108, use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound.

In Exercises 107 and 108, use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound.

In Exercises 109114, evaluate the expression. If necessary, round your answer to two decimal places.

In Exercises 109114, evaluate the expression. If necessary, round your answer to two decimal places.

In Exercises 109114, evaluate the expression. If necessary, round your answer to two decimal places.

In Exercises 109114, evaluate the expression. If necessary, round your answer to two decimal places.

In Exercises 109114, evaluate the expression. If necessary, round your answer to two decimal places.

In Exercises 109114, evaluate the expression. If necessary, round your answer to two decimal places.

In Exercises 115118, evaluate the expression without the aid of a calculator.

In Exercises 115118, evaluate the expression without the aid of a calculator.

In Exercises 115118, evaluate the expression without the aid of a calculator.

In Exercises 115118, evaluate the expression without the aid of a calculator.

In Exercises 119122, use a calculator to evaluate the expression. Round your answer to two decimal places.

In Exercises 119122, use a calculator to evaluate the expression. Round your answer to two decimal places.

In Exercises 119122, use a calculator to evaluate the expression. Round your answer to two decimal places.

In Exercises 119122, use a calculator to evaluate the expression. Round your answer to two decimal places.

In Exercises 123126, use a graphing utility to graph the function.

In Exercises 123126, use a graphing utility to graph the function.

In Exercises 123126, use a graphing utility to graph the function.

In Exercises 123126, use a graphing utility to graph the function.

In Exercises 127130, find the exact value of the expression.

In Exercises 127130, find the exact value of the expression.

In Exercises 127130, find the exact value of the expression.

In Exercises 127130, find the exact value of the expression.

In Exercises 131 and 132, write an algebraic expression that is equivalent to the expression.

In Exercises 131 and 132, write an algebraic expression that is equivalent to the expression.

Angle of Elevation The height of a radio transmission tower is 70 meters, and it casts a shadow of length 30 meters (see figure). Find the angle of elevation of the sun. 13

Height Your football has landed at the edge of the roof of your school building. When you are 25 feet from the base of the building, the angle of elevation to your football is How high off the ground is your football? 13

Distance From city to city a plane flies 650 miles at a bearing of From city to city the plane flies 810 miles at a bearing of . Find the distance from city A to city C and the bearing from city A to city C. 13

Wave Motion Your fishing bobber oscillates in simple harmonic motion from the waves in the lake where you fish. Your bobber moves a total of 1.5 inches from its high point to its low point and returns to its high point every 3 seconds. Write an equation modeling the motion of your bobber if it is at its high point at time True or False? 13

The tangent function is often useful for modeling simple harmonic motion. 13

The inverse sine function arcsin cannot be defined as a function over any interval that is greater than the interval defined as 13

is not a function because 14

Because tan 14

In Exercises 141144, match the function with its graph. Base your selection solely on your interpretation of the constants and Explain your reasoning. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 141144, match the function with its graph. Base your selection solely on your interpretation of the constants and Explain your reasoning. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 141144, match the function with its graph. Base your selection solely on your interpretation of the constants and Explain your reasoning. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 141144, match the function with its graph. Base your selection solely on your interpretation of the constants and Explain your reasoning. [The graphs are labeled (a), (b), (c), and (d).]

Writing Describe the behavior of at the zeros of Explain your reasoning. 14

Conjecture (a) Use a graphing utility to complete the table. (b) Make a conjecture about the relationship between and 14

Writing When graphing the sine and cosine functions, determining the amplitude is part of the analysis. Explain why this is not true for the other four trigonometric functions. 14

Oscillation of a Spring A weight is suspended from a ceiling by a steel spring. The weight is lifted (positive direction) from the equilibrium position and released. The resulting motion of the weight is modeled by where is the distance in feet from equilibrium and is the time in seconds. The graph of the function is shown in the figure. For each of the following, describe the change in the system without graphing the resulting function. (a) is changed from to (b) is changed from to (c) is changed from 6 to 9. 14

Graphical Reasoning The formulas for the area of a circular sector and arc length are and respectively. ( is the radius and is the angle measured in radians.) (a) For write the area and arc length as functions of What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more rapidly as increases. Explain. (b) For centimeters, write the area and arc length as functions of What is the domain of each function? Use a graphing utility to graph and identify the functions. 15

Writing Describe a real-life application that can be represented by a simple harmonic motion model and is different from any that youve seen in this chapter. Explain which function you would use to model your application and why. Explain how you would determine the amplitude, period, and frequency of the model for your application.

Consider an angle that measures radians. (a) Sketch the angle in standard position. (b) Determine two coterminal angles (one positive and one negative). (c) Convert the angle to degree measure.

A truck is moving at a rate of 90 kilometers per hour, and the diameter of its wheels is 1 meter. Find the angular speed of the wheels in radians per minute.

A water sprinkler sprays water on a lawn over a distance of 25 feet and rotates through an angle of Find the area of the lawn watered by the sprinkler.

Find the exact values of the six trigonometric functions of the angle shown in the figure.

Given that find the other five trigonometric functions of

Determine the reference angle of the angle and sketch and in standard position.

Determine the quadrant in which lies if and

Find two exact values of in degrees if (Do not use a calculator.)

Use a calculator to approximate two values of in radians if Round the results to two decimal places. 1

In Exercises 10 and 11, find the remaining five trigonometric functions of satisfying the conditions.

In Exercises 10 and 11, find the remaining five trigonometric functions of satisfying the conditions.

In Exercises 12 and 13, sketch the graph of the function. (Include two full periods.)

In Exercises 12 and 13, sketch the graph of the function. (Include two full periods.)

In Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic, find its period.

In Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic, find its period.

Find and for the function such that the graph of matches the figure.

Find the exact value of without the aid of a calculator.

Graph the function

A plane is 80 miles south and 95 miles east of Cleveland Hopkins International Airport. What bearing should be taken to fly directly to the airport? 2

Write the equation for the simple harmonic motion of a ball on a spring that starts at its lowest point of 6 inches below equilibrium, bounces to its maximum height of 6 inches above equilibrium, and returns to its lowest point in a total of 2 seconds.

The restaurant at the top of the Space Needle in Seattle, Washington is circular and has a radius of 47.25 feet. The dining part of the restaurant revolves, making about one complete revolution every 48 minutes. A dinner party was seated at the edge of the revolving restaurant at 6:45 P.M. and was finished at 8:57 P.M. (a) Find the angle through which the dinner party rotated. (b) Find the distance the party traveled during dinner.

A bicycles gear ratio is the number of times the freewheel turns for every one turn of the chainwheel (see figure). The table shows the numbers of teeth in the freewheel and chainwheel for the first five gears of an 18-speed touring bicycle. The chainwheel completes one rotation for each gear. Find the angle through which the freewheel turns for each gear. Give your answers in both degrees and radians.

A surveyor in a helicopter is trying to determine the width of an island, as shown in the figure. (a) What is the shortest distance the helicopter would have to travel to land on the island? (b) What is the horizontal distance that the helicopter would have to travel before it would be directly over the nearer end of the island? (c) Find the width of the island. Explain how you obtained your answer.

Use the figure below. (a) Explain why are similar triangles. (b) What does similarity imply about the ratios (c) Does the value of sin depend on which triangle from part (a) is used to calculate it? Would the value of sin change if it were found using a different right triangle that was similar to the three given triangles? (d) Do your conclusions from part (c) apply to the other five trigonometric functions? Explain.

Use a graphing utility to graph and use the graph to decide whether is even, odd, or neither. (a) (b)

If is an even function and is an odd function, use the results of Exercise 5 to make a conjecture about where (a) (b)

The model for the height (in feet) of a Ferris wheel car is where is the time (in minutes). (The Ferris wheel has a radius of 50 feet.) This model yields a height of 50 feet when Alter the model so that the height of the car is 1 foot when t _ 0.

The pressure (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by where is time (in seconds). (a) Use a graphing utility to graph the model. (b) What is the period of the model? What does the period tell you about this situation? (c) What is the amplitude of the model? What does it tell you about this situation? (d) If one cycle of this model is equivalent to one heartbeat, what is the pulse of this patient? (e) If a physician wants this patients pulse rate to be 64 beats per minute or less, what should the period be? What should the coefficient of be?

A popular theory that attempts to explain the ups and downs of everyday life states that each of us has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by sine waves. Physical (23 days): Emotional (28 days): Intellectual (33 days): where is the number of days since birth. Consider a person who was born on July 20, 1986. (a) Use a graphing utility to graph the three models in the same viewing window for (b) Describe the persons biorhythms during the month of September 2006. (c) Calculate the persons three energy levels on September 22, 2006. 1

(a) Use a graphing utility to graph the functions given by and (b) Use the graphs from part (a) to find the period of each function. (c) If are positive integers, is the function given by periodic? Explain your reasoning. 1

Two trigonometric functions and have periods of 2, and their graphs intersect at (a) Give one smaller and one larger positive value of at which the functions have the same value. (b) Determine one negative value of at which the graphs intersect. (c) Is it true that Explain your reasoning. 1

The function is periodic, with period So, Are the following equal? Explain. (a) (b) (c) 1

If you stand in shallow water and look at an object below the surface of the water, the object will look farther away from you than it really is. This is because when light rays pass between air and water, the water refracts, or bends, the light rays. The index of refraction for water is 1.333. This is the ratio of the sine of and the sine of (see figure). (a) You are standing in water that is 2 feet deep and are looking at a rock at angle (measured from a line perpendicular to the surface of the water). Find (b) Find the distances and (c) Find the distance between where the rock is and where it appears to be. (d) What happens to as you move closer to the rock? Explain your reasoning. 1

In calculus, it can be shown that the arctangent function can be approximated by the polynomial where is in radians. (a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Study the pattern in the polynomial approximation of the arctangent function and guess the next term. Then repeat part (a). How does the accuracy of the approximation change when additional terms are added?