4.8 Solving Problems with Trigonometry

More Right Triangle Problems

We close this first of two trigonometry chapters by revisiting some of the applications of Section 4.2 (right triangle trigonometry) and Section 4.4 (sinusoids).

An angle of elevation is the angle through which the eye moves up from horizontal to look at something above, and an angle of depression is the angle through which the eye moves down from horizontal to look at something below. For two observers at different elevations looking at each other, the angle of elevation for one equals the angle of depression for the other. The concepts are illustrated in Figure 4.91 as they might apply to observers at Mount Rushmore or the Grand Canyon.

What you’ll learn about

tMore Right Triangle Problems tSimple Harmonic Motion

... and why

These problems illustrate some of the better-known applications of trigonometry.

4.8 Solving Problems with Trigonometry

FIGURE 4.91 (a) Angle of elevation at Mount Rushmore. (b) Angle of depression at the Grand Canyon.

Line ofsight

Angle of elevation

Line ofsight

B

A Angle of depression

(a) (b)

FIGURE 4.92 A big lighthouse and a little buoy. (Example 1)

6°

130

u x

EXAMPLE 1

Using Angle of Depression

The angle of depression of a buoy from the top of the Barnegat Bay lighthouse 130 ft above the surface of the water is 6 °. Find the distance x from the base of the lighthouse to the buoy.

SOLUTION

Figure 4.92 models the situation.

In the diagram, u = 6° because the angle of elevation from the buoy equals the angle of depression from the lighthouse. We solve algebraically using the tangent function:

tan u = tan 6° = 130

x = 130

tan 6 ° ≈ 1236.9

Interpreting We find that the buoy is about 1237 ft from the base of the lighthouse.

Now try Exercise 3.

SECTION 4.8 Solving Problems with Trigonometry 389

EXAMPLE 2

Making Indirect Measurements

From the top of the 100-ft-tall Altgelt Hall a man observes a car moving toward the building. If the angle of depression of the car changes from 22 ° to 46° during the period of observation, how far does the car travel?

SOLUTION

Solve Algebraically Figure 4.93 models the situation. Notice that we have labeled

the acute angles at the car’s two positions as 22 ° and 46° (because the angle of elevation from the car equals the angle of depression from the building). Denote the distance the car moves as x. Denote its distance from the building at the second observation as d.

From the smaller right triangle we conclude:

tan 46 ° = 100

d = 100

tan 46 ° From the larger right triangle we conclude:

tan 22 ° = 100

+ d x + d = 100

tan 22 ° x = 100

tan 22 ° - d x = 100

tan 22 ° - 100 tan 46 ° x ≈ 150.9

Interpreting We find that the car travels about 151 ft. Now try Exercise 7.

FIGURE 4.93 A car approaches Altgelt Hall. (Example 2)

22° 46°

x d

100 ft 22°

46°

10 ft

FIGURE 4.94 A large, helium-filled penguin. (Example 3)

x

h

40° 48° 10

FIGURE 4.95 (Example 3)

EXAMPLE 3

Finding Height Above Ground

A large, helium-filled penguin is moored at the beginning of a parade route await- ing the start of the parade. Two cables attached to the underside of the penguin make angles of 48 ° and 40° with the ground and are in the same plane as a perpen- dicular line from the penguin to the ground. (See Figure 4.94.) If the cables are attached to the ground 10 ft from each other, how high above the ground is the penguin?

SOLUTION

We can simplify the drawing to the two right triangles in Figure 4.95 that share the common side h.

Model By the definition of the tangent function, h

x

= tan 48° and h

x

+ 10 = tan 40°.

Solve Algebraically Solving for h,

h

= x tan 48° and h = 1x + 102 tan 40°.

(continued )

Set these two expressions for h equal to each other and solve the equation for x:

x tan 48 ° = 1x + 102 tan 40°

x tan 48 ° = x tan 40° + 10 tan 40°

x tan 48 ° - x tan 40° = 10 tan 40°

x 1tan 48° - tan 40°2 = 10 tan 40°

x = 10 tan 40 °

tan 48 ° - tan 40° ≈ 30.90459723

We retain the full display for x because we are not finished yet; we need to solve for h:

h

= x tan 48° = 130.904597232 tan 48° ≈ 34.32 The penguin is approximately 34 ft above ground level.

Now try Exercise 15.

FIGURE 4.96 Path of travel for a Coast Guard boat that corners well at 35 knots.

(Example 4)

North 53° 143°

β α A θ

B

C

Initial (t 5 0) position

2a d a 1d

0 Piston

avt

FIGURE 4.97 A piston operated by a wheel rotating at a constant rate demonstrates simple harmonic motion.

EXAMPLE 4

Using Trigonometry in Navigation

A U.S. Coast Guard patrol boat leaves Port Cleveland and averages 35 knots (naut mi per hr) traveling for 2 hr on a course of 53 ° and then 3 hr on a course of 143°.

What is the boat’s bearing and distance from Port Cleveland?

SOLUTION

Figure 4.96 models the situation.

Solve Algebraically In the diagram, line AB is a transversal that cuts a pair of paral-

lel lines. Thus, b = 53° because they are alternate interior angles. Angle a, as the supplement of a 143 ° angle, is 37°. Consequently, ∠ABC = 90° and AC is the hypotenuse of right △ ABC.

Use distance = rate * time to determine distances AB and BC.

AB = 135 knots212 hr2 = 70 naut mi BC = 135 knots213 hr2 = 105 naut mi Solve the right triangle for AC and u.

AC = 270

+ 105

AC ≈ 126.2 u = tan

a 105

70 b u ≈ 56.3°

Interpreting We find that the boat’s bearing from Port Cleveland is 53

° + u, or ap- proximately 109.3 °. They are about 126 naut mi out.

Now try Exercise 17.

Simple Harmonic Motion

Because of their periodic nature, the sine and cosine functions are helpful in describing the motion of objects that oscillate, vibrate, or rotate. For example, the linkage in Figure 4.97 converts the rotary motion of a motor to the back-and-forth motion needed for some machines. When the wheel rotates, the piston moves back and forth.

If the wheel rotates at a constant rate v radians per second, the back-and-forth motion of the piston is an example of simple harmonic motion and can be modeled by an equa- tion of the form

d

= a cos vt, v 7 0,

where a is the radius of the wheel and d is the directed distance of the piston from its

center of oscillation.

SECTION 4.8 Solving Problems with Trigonometry 391

For the sake of simplicity, we will define simple harmonic motion in terms of a point moving along a number line.

Frequency and Period

Notice that harmonic motion is sinusoidal, with amplitude 0a0 and period 2p/v. The frequency is the reciprocal of the period.

DEFINITION Simple Harmonic Motion

A point moving on a number line is in simple harmonic motion if its directed distance d from the origin is given by either

d

= a sin vt or d = a cos vt,

where a and v are real numbers and v 7 0. The motion has frequency v/2p, which is the number of oscillations per unit of time.

EXPLORATION 1 Watching Harmonic Motion

You can watch harmonic motion on your graphing calculator. Set your grapher to parametric mode and set X

= cos 1T2 and Y

= sin 1T2. Set Tmin = 0, Tmax = 25, Tstep = 0.2, Xmin = -1.5, Xmax = 1.5, Xscl = 1, Ymin = -100, Ymax = 100, Yscl = 0.

If your calculator allows you to change style to graph a moving ball, choose that style. When you graph the function, you will see the ball moving along the x-axis between -1 and 1 in simple harmonic motion. If your grapher does not have the moving ball option, wait for the grapher to finish graphing, then press TRACE and keep your finger pressed on the right arrow key to see the tracer move in simple harmonic motion.

1.

For each value of T, the parametrization gives the point 1cos 1T2, sin 1T22.

What well-known curve should this parametrization produce?

2.

Why does the point seem to go back and forth on the x-axis when it should be following the curve identified in part 1? (Hint: Check that viewing window again!)

3.

Why does the point slow down at the extremes and speed up in the middle?

(Hint: Remember that the grapher is really following the curve identified in part 1.)

4.

How can you tell that this point moves in simple harmonic motion?

FIGURE 4.98 Modeling the path of a piston by a sinusoid. (Example 5)

y

x 8

P

t 8 cos 8_π

t 8_π

EXAMPLE 5

Calculating Harmonic Motion

In a mechanical linkage like the one shown in Figure 4.97, a wheel with an 8-cm radius turns with an angular velocity of 8 p rad/sec.

(a) What is the frequency of the piston in cycles per second?

(b) What is the distance from the starting position 1t = 02 exactly 3.45 sec after

starting?

SOLUTION

Imagine the wheel to be centered at the origin and let P 1x, y2 be a point on its perimeter (Figure 4.98). As the wheel rotates and P goes around, the motion of the piston follows the path of the x-coordinate of P along the x-axis. The angle determined by P at any time t is 8 pt, so its x-coordinate is 8 cos 8pt. Therefore, the sinusoid d = 8 cos 8pt models the motion of the piston.

(continued )

(a) The frequency of d

= 8 cos 8pt is 8p/2p, or 4 cycles/sec. The piston makes four complete back-and-forth strokes per second. The graph of d as a function of

30, 14 model the four cycles of the motor or the four strokes of the piston. Note that the sinusoid has a period of 1/4, the reciprocal of the frequency.

(b) We must find the distance between the positions at t

= 0 and t = 3.45.

The initial position at t = 0 is

d

102 = 8.

The position at t = 3.45 is

d

13.452 = 8 cos 18p # 3.45 2 ≈ 2.47.

The distance between the two positions is approximately 8 - 2.47 = 5.53.

Interpreting We conclude that the piston is approximately 5.53 cm from its starting

position after 3.45 sec.

FIGURE 4.99 A sinusoid with frequency 4 models the motion of the piston in Example 5.

[0, 1] by [–10, 10]

EXAMPLE 6

Calculating Harmonic Motion

A mass oscillating up and down on the bottom of a spring (assuming perfect elastic- ity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 5 cm, find the modeling equation if it takes 2 sec to complete one cycle. (See Figure 4.100.)

5 cm

0 cm

⫺5 cm

5 cm

0 cm

⫺5 cm

FIGURE 4.100 The mass and spring in Example 6.

SOLUTION

We have our choice between the two equations d = a sin vt or

= a cos vt. Assuming that the spring is at the origin of the coordinate system when t = 0, we choose the equation d = a sin vt.

Because the maximum displacement is 5 cm, we conclude that the amplitude a = 5.

Because it takes 2 sec to complete one cycle, we conclude that the period is 2 and the frequency is 1/2. Therefore,

v 2 p = 1

2 v = p

Putting it all together, our modeling equation is d = 5 sin pt.

Now try Exercise 29.

SECTION 4.8 Solving Problems with Trigonometry 393

CHAPTER OPENER Problem (from page 319)

Problem: If we know that the musical note A above middle C has a pitch of

440 Hz, how can we model the sound produced by it at 80 dB?

Solution: Sound is modeled by simple harmonic motion, with frequency per-

ceived as pitch and measured in hertz (Hz), and amplitude perceived as loudness and measured in decibels (dB). So for the musical note A with a pitch of 440 Hz, we have frequency = v/2p = 440 and therefore v = 2p440 = 880p.

If this note is played at a loudness of 80 dB, we have 0

0 = 80. Using the simple harmonic motion model d = a sin vt, we have

d

= 80 sin 880pt.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

In Exercises 1–4, find the lengths a, b, and c.

1. 2.

3. 4.

In Exercises 5 and 6, find the complement and supplement of the angle.

5. 32° 6. 73°

In Exercises 7 and 8, state the bearing that describes the direction.

7. NE (northeast) 8. SSW (south-southwest)

In Exercises 9 and 10, state the amplitude and period of the sinusoid.

9. -3 sin 21x - 12 10. 4 cos 41x + 22

5. 6.

7.

8.

9.

10.

QUICK REVIEW 4.8

15

31° c

b

68° 25 a

b

28

44° 28° a

c

b

21

48° 31° a

c

b

2. Finding a Monument Height From a point 100 ft from its base, the angle of elevation of the top of the Arch of Septimus Severus, in Rome, Italy, is 34°13′12″. How tall is this monument?

3. Finding a Distance The angle of depression from the top of the Smoketown Lighthouse 120 ft above the surface of the water to a buoy is 10°. How far is the buoy from the lighthouse?

10° 120 ft

SECTION 4.8 Exercises

In Exercises 1– 43, solve the problem using your knowledge of geom- etry and the techniques of this section. Sketch a figure if one is not provided.

1. Finding a Cathedral Height The angle of eleva- tion of the top of the Ulm Cathedral from a point 300 ft away from the base of its stee- ple on level ground is 60°.

Find the height of the cathedral.

300 ft 60° h

11. Antenna Height A guy wire attached to the top of the KSAM radio antenna is anchored at a point on the ground 10 m from the antenna’s base. If the wire makes an angle of 55° with level ground, how high is the KSAM antenna?

12. Building Height To determine the height of the Louisiana- Pacific (LP) Tower, the tallest building in Conroe, Texas, a sur- veyor stands at a point on the ground, level with the base of the LP building. He measures the point to be 125 ft from the building’s base and the angle of elevation to the top of the building to be 29°48′. Find the height of the building.

13. Navigation The Paz Verde, a whalewatch boat, is located at point P, and L is the nearest point on the Baja California shore. Point Q is located 4.25 mi down the shoreline from L and PL # LQ. Determine the distance that the Paz Verde is from the shore if ∠PQL = 35°.

4. Finding a Baseball Stadium Dimension The top row of seats behind home plate at Cincinnati’s Great American Ball Park is 90 ft above the level of the playing field. The angle of depression to the base of the left field wall is 14°. How far is the base of the left field wall from a point on level ground directly below the top row?

5. Finding a Guy-Wire Length A guy wire connects the top of an antenna to a point on level ground 50 ft from the base of the antenna. The angle of elevation formed by this wire is 80° . What are the length of the wire and the height of the antenna?

6. Finding a Length A wire stretches from the top of a ver- tical pole to a point on level ground 16 ft from the base of the pole. If the wire makes an angle of 62° with the ground, find the height of the pole and the length of the wire.

7. Height of Eiffel Tower The angle of elevation of the top of the TV antenna mounted on top of the Eiffel Tower in Paris is measured to be 80°1′12″ at a point 185 ft from the base of the tower. How tall is the tower plus TV antenna?

8. Finding the Height of Tallest Chimney The world’s tallest smokestack at the International Nickel Co., Sudbury, Ontario, casts a shadow that is approximately 1580 ft long when the Sun’s angle of elevation (measured from the horizon) is 38°. How tall is the smokestack?

Sun

Smokestack

Shadow ⫽ 1580 ft 38°

38°

9. Cloud Height To measure the height of a cloud, you place a bright searchlight directly below the cloud and shine the beam straight up. From a point 100 ft away from the search- light, you measure the angle of elevation of the cloud to be 83°12′. How high is the cloud?

10. Ramping Up A ramp leading to a freeway overpass is 470 ft long and rises 32 ft. What is the average angle of inclination of the ramp to the nearest tenth of a degree?

L

P

35°

4.25 mi Q

200 ft 40°

30° 50 ft

80°

14. Recreational Hiking While hiking on a level path to- ward Colorado’s front range, Otis Evans determines that the angle of elevation to the top of Long’s Peak is 30°. Moving 1000 ft closer to the mountain, Otis determines the angle of elevation to be 35°. How much higher is the top of Long’s Peak than Otis’s elevation?

15. Civil Engineering The angle of elevation from an ob- server to the bottom edge of the Delaware River drawbridge observation deck located 200 ft from the observer is 30°. The angle of elevation from the observer to the top of the observa- tion deck is 40°. What is the height of the observation deck?

16. Traveling Car From the top of a 100-ft building a man observes a car moving toward him. If the angle of depression of the car changes from 15° to 33° during the period of obser- vation, how far does the car travel?

33° 100 ft

15°

SECTION 4.8 Solving Problems with Trigonometry 395

24. Recreational Flying A hot-air balloon over Park City, Utah, is 760 ft above the ground. The angle of depression from the balloon to an observer is 5.25°. Assuming the ground is rel- atively flat, how far is the observer from a point on the ground directly under the balloon?

25. Navigation A shoreline runs north-south, and a boat is due east of the shoreline. The bearings of the boat from two points on the shore are 110° and 100°. Assume the two points are 550 ft apart. How far is the boat from the shore?

17. Navigation The Coast Guard cutter Angelica travels at 30 knots from its home port of Corpus Christi on a course of 95° for 2 hr and then changes to a course of 185° for 2 hr.

Find the distance and the bearing from the Corpus Christi port to the boat.

18. Navigation The Cerrito Lindo travels at a speed of 40 knots from Fort Lauderdale on a course of 65° for 2 hr and then changes to a course of 155° for 4 hr. Determine the distance and the bearing from Fort Lauderdale to the boat.

19. Land Measure The angle of depression is 19° from a point 7256 ft above sea level on the north rim of the Grand Canyon level to a point 6159 ft above sea level on the south rim. How wide is the canyon at that point?

20. Ranger Fire Watch A ranger spots a fire from a 73-ft tower in Yellowstone National Park. She measures the angle of depression to be 1°20′. How far is the fire from the tower?

21. Civil Engineering The bearing of the line of sight to the east end of the Royal Gorge footbridge from a point 375 ft due north of the west end of the footbridge across the Royal Gorge is 113°. What is the length l of the bridge?

Corpus Christi

95°

185°

375 ft

l 113°

12

h 17°

73 ft 15° 15°

22. Space Flight The angle of elevation of a space shuttle from Cape Canaveral is 17° when the shuttle is directly over a ship 12 mi downrange. What is the altitude of the shuttle when it is directly over the ship?

23. Architectural Design A barn roof is constructed as shown in the figure at the top of the next column. What is the height of the vertical center span?

26. Navigation Milwaukee, Wisconsin, is directly west of Grand Haven, Michigan, on opposite sides of Lake Michigan.

On a foggy night, a law enforcement boat leaves from Milwaukee on a course of 105° at the same time that a small smuggling craft steers a course of 195° from Grand Haven.

The law enforcement boat averages 23 knots and collides with the smuggling craft. What was the smuggling boat’s average speed?

27. Mechanical Design Refer to Figure 4.97. The wheel in a piston linkage like the one shown in the figure has a radius of 6 in. It turns with an angular velocity of 16p rad/sec. The initial position is the same as that shown in Figure 4.97.

(a) What is the frequency of the piston?

(b) What equation models the motion of the piston?

(c) What is the distance from the initial position 2.85 sec after starting?

28. Mechanical Design Suppose the wheel in a piston link- age like the one shown in Figure 4.97 has a radius of 18 cm and turns with an angular velocity of p rad/sec.

(a) What is the frequency of the piston?

(b) What equation models the motion of the piston?

(c) How many cycles does the piston make in 1 min?

550 ft 110°

100°

(e) Use your sinusoidal model to predict dates in the year when the mean temperature in Charleston will be 70°.

(Assume that t = 0 represents January 1.) 29. Vibrating Spring A mass on a spring

oscillates back and forth and completes one cycle in 0.5 sec. Its maximum dis- placement is 3 cm. Write an equation that models this motion.

30. Tuning Fork A point on the tip of a tun- ing fork vibrates in harmonic motion described by the equation d = 14 sin vt. Find v for a tuning fork that has a frequency of 528 vibra- tions per second.

31. Ferris Wheel Motion The Ferris wheel shown in this figure makes one complete turn every 20 sec. A rider’s height, h, above the ground can be modeled by the equation

h = a sin vt + k, where h and k are given in feet and t is given in seconds.

0 1

10 20 30 40 50 60 70 80 90

2 3 4 5

Time (months)

Temperature (°F)

6 7 8 9 10 11 12 x y

0 cm

d cm

8 ft

25 ft

(a) What is the value of a?

(b) What is the value of k?

(c) What is the value of v?

32. Ferris Wheel Motion Jacob and Emily ride a Ferris wheel at a carnival in Billings, Montana. The wheel has a 16-m diameter and turns at 3 rpm with its lowest point 1 m above the ground. Assume that Jacob and Emily’s height h above the ground is a sinusoidal function of time t (in seconds), where t = 0 represents the lowest point of the wheel.

(a) Write an equation for h.

(b) Draw a graph of h for 0 … t … 30.

(c) Use h to estimate Jacob and Emily’s height above the ground at t = 4 and t = 10.

33. Monthly Temperatures in Charleston The monthly normal mean temperatures in Charleston, SC, are shown in Table 4.3. A scatter plot suggests that the mean monthly tem- peratures follow a sinusoidal curve over time. Assume that the sinusoid has equation y = a sin 1b 1t - h22 + k.

(a) Given that the period is 12 months, find b.

(b) Assuming that the high and low temperatures in the table determine the range of the sinusoid, find a and k.

(c) Find a value of h that will put the minimum at t= 1 and the maximum at t = 7.

(d) Superimpose a graph of your sinusoid on a scatter plot of the data. How good is the fit?

Table 4.3 Average Daily Temperature for Charleston, SC

Source: Southeast Regional Climate Center, 2013.

Month Temperature (°F)

1 49.2

2 51.3

3 57.7

4 64.9

5 72.7

6 78.8

7 81.5

8 80.7

9 76.3

10 66.8

11 57.9

12 50.9

34. Writing to Learn For the Ferris wheel in Exercise 31, which equation correctly models the height of a rider who be- gins the ride at the bottom of the wheel when t = 0?

(a) h = 25 sin pt 10 (b) h = 25 sin pt

10 + 8 (c) h = 25 sin pt

10 + 33 (d) h = 25 sin apt

10 + 3p 2b + 33

Explain your thought process, and use of a graphing utility in choosing the correct modeling equation.

SECTION 4.8 Solving Problems with Trigonometry 397

42. Multiple Choice The loudness of a musical tone is determined by which characteristic of its sound wave?

(A) Amplitude (B) Frequency (C) Period (D) Phase shift (E) Pitch

Explorations

43. Group Activity The data for displacement versus time on a tuning fork, shown in Table 4.4, were collected using a CBL and a microphone.

35. Monthly Sales Owing to startup costs and seasonal varia- tions, Gina found that the monthly profit in her bagel shop dur- ing the first year followed an up-and-down pattern that could be modeled by P = 2t - 7 sin 1pt/32, where P was measured in hundreds of dollars and t was measured in months after January 1.

(a) In what month did the shop first begin to make money?

(b) In what month did the shop enjoy its greatest profit in that first year?

36. Weight Loss Courtney tried several different diets over a two-year period in an attempt to lose weight. She found that her weight W followed a fluctuating curve that could be mod- eled by the function W = 220 - 1.5t + 9.81 sin 1pt/42, where t was measured in months after January 1 of the first year and W was measured in pounds.

(a) What was Courtney’s weight at the start and at the end of two years?

(b) What was her maximum weight during the two-year period?

(c) What was her minimum weight during the two-year period?

Standardized Test Questions

37. True or False Higher frequency sound waves have shorter periods. Justify your answer.

38. True or False A car traveling at 30 mph is traveling faster than a ship traveling at 30 knots. Justify your answer.

You may use a graphing calculator when answering these questions.

39. Multiple Choice To get a rough idea of the height of a building, John paces off 50 ft from the base of the building, then measures the angle of elevation from the ground to the top of the building at that point to be 58°. About how tall is the building?

(A) 31 ft (B) 42 ft (C) 59 ft (D) 80 ft (E) 417 ft

40. Multiple Choice A boat leaves harbor and travels at 20 knots on a bearing of 90°. After 2 hr, it changes course to a bearing of 150° and continues at the same speed for another hour.

After the entire 3-hr trip, how far is it from the harbor?

(A) 50 naut mi (B) 53 naut mi (C) 57 naut mi (D) 60 naut mi (E) 67 naut mi

41. Multiple Choice At high tide at 8:15 p.m., the water level on the side of a pier is 9 ft from the top. At low tide 6 hr 12 min later, the water level is 13 ft from the top. At which of the following times in that interval is the water level 10 ft from the top of the pier?

(A) 9:15 p.m. (B) 9:48 p.m. (C) 9:52 p.m.

(d) 10:19 p.m. (E) 11:21 p.m.

Table 4.4 Tuning Fork Data

Time Displacement Time Displacement

0.00091 -0.080 0.00362 0.217

0.00108 0.200 0.00379 0.480

0.00125 0.480 0.00398 0.681

0.00144 0.693 0.00416 0.810

0.00162 0.816 0.00435 0.827

0.00180 0.844 0.00453 0.749

0.00198 0.771 0.00471 0.581

0.00216 0.603 0.00489 0.346

0.00234 0.368 0.00507 0.077

0.00253 0.099 0.00525 -0.164

0.00271 -0.141 0.00543 -0.320

0.00289 -0.309 0.00562 -0.354

0.00307 -0.348 0.00579 -0.248

0.00325 -0.248 0.00598 -0.035

0.00344 -0.041    

(a) Graph a scatter plot of the data in the 30, 0.00624 by 3-0.5, 14 viewing window.

(b) Select the equation that appears to be the best fit of these data.

i. y = 0.6 sin 12464x - 2.842 + 0.25 ii. y = 0.6 sin 11210x - 22 + 0.25 iii. y = 0.6 sin 12440x - 2.12 + 0.15

(c) What is the approximate frequency of the tuning fork?

44. Writing to Learn Human sleep-awake cycles at three dif- ferent ages are described by the accompanying graphs. The portions of the graphs above the horizontal lines represent times awake, and the portions below represent times asleep.

6 P.M. 12 6 A.M.

Newborn

12 6 P.M.

6 P.M. 12 6 A.M.

Four years

12 6 P.M.

6 P.M. 12 6 A.M.

Adult

12 6 P.M.

51. Group Activity A musical note like that produced with a tuning fork or pitch meter is a pressure wave. Typically, fre- quency is measured in hertz 11 Hz = 1 cycle per second2.

Table 4.5 gives frequency (in Hz) of several musical notes. The time-vs.-pressure tuning fork data in Table 4.6 was collected using a CBL and a microphone.

(a) What is the period of the sleep-awake cycle of a newborn?

of a four-year-old? of an adult?

(b) Which of these three sleep-awake cycles is the closest to being modeled by a function y = a sin bx?

Using Trigonometry in Geometry In a regular polygon all sides have equal length and all angles have equal measure. In Exercises 45 and 46, consider the regular seven-sided polygon whose sides are 5 cm.

45. Find the length of the apothem, the segment from the center of the seven-sided polygon to the midpoint of a side.

46. Find the radius of the circumscribed circle of the regular seven- sided polygon.

47. A rhombus is a quadrilateral with all sides equal in length. Recall that a rhombus is also a parallelogram.

Find length AC and length BD in the rhombus shown here.

Extending the Ideas

48. A roof has two sections, one with a 50° elevation and the other with a 20° elevation, as shown in the figure.

(a) Find the height BE.

(b) Find the height CD.

(c) Find the length AE + ED, and double it to find the length of the roofline.

5 cm

a r

18 in.

C A

D B 42°

C A

D E

B 20 ft 45 ft

50° 50°

20° 20°

49. Steep Trucking The percentage grade of a road is its slope expressed as a percentage. A tractor-trailer rig passes a sign that reads, “6% grade next 7 miles.” What is the average angle of inclination of the road?

50. Television Coverage Many satellites travel in geosyn- chronous orbits, which means that the satellite stays over the same point on the Earth. A satellite that broadcasts cable television is in geosynchronous orbit 100 mi above the Earth.

Assume that the Earth is a sphere with radius 4000 mi, and find the arc length of coverage area for the cable television satellite on the Earth’s surface.

Table 4.6 Tuning Fork Data

Time (sec) Pressure Time (sec) Pressure 0.0002368 1.29021 0.0049024 -1.06632 0.0005664 1.50851 0.0051520 0.09235 0.0008256 1.51971 0.0054112 1.44694 0.0010752 1.51411 0.0056608 1.51411 0.0013344 1.47493 0.0059200 1.51971 0.0015840 0.45619 0.0061696 1.51411 0.0018432 -0.89280 0.0064288 1.43015 0.0020928 -1.51412 0.0066784 0.19871 0.0023520 -1.15588 0.0069408 -1.06072 0.0026016 -0.04758 0.0071904 -1.51412 0.0028640 1.36858 0.0074496 -0.97116 0.0031136 1.50851 0.0076992 0.23229 0.0033728 1.51971 0.0079584 1.46933 0.0036224 1.51411 0.0082080 1.51411 0.0038816 1.45813 0.0084672 1.51971 0.0041312 0.32185 0.0087168 1.50851 0.0043904 -0.97676 0.0089792 1.36298

0.0046400 -1.51971    

Table 4.5 Tuning Fork Data

Note Frequency (Hz)

C 262

C♯ or D♭ 277

D 294

D♯ or E♭ 311

E 330

F 349

F ♯ or G♭ 370

G 392

G♯ or A♭ 415

A 440

A♯ or B♭ 466

B 494

C (next octave) 524

(a) Graph a scatter plot of the data.

(b) Determine a, b, and h so that the equation y = a sin 1b1t - h22 is a model for the data.

(c) Determine the frequency of the sinusoid in part (b), and use Table 4.5 to identify the musical note produced by the tuning fork.

(d) Identify the musical note produced by the tuning fork used in Exercise 43.

CHAPTER 4 Key Ideas

Properties, Theorems, and Formulas Arc Length Formulas 322, 323

Distance Conversions 324 Special Angles 330, 331, 346 Period of a Sinusoid 353 Frequency of a Sinusoid 353

Graphs of Sinusoids 354

Sums That Are Sinusoidal Functions 371 Procedures

Degree-Radian Conversion 322

Evaluating Trig Functions of a Nonquadrantal Angle 341 Constructing a Sinusoidal Model Using Time 355

f(x) ⫽ sin x by [–4, 4]

2 _π] , [–2_π

f(x) ,2 _π⫽ cos x]by [–4, 4]

[–2π [23p/2, 3p/2] by [–4, 4]

f(x) 5 tan x

f(x) ⫽ cot x by [–4, 4]

2 _π] ,

[–2_π [–2_π,2 _π]by [–4, 4]

f(x) ⫽ sec x

by [–4, 4]

2 _π] , [–2_π

f(x) ⫽ csc x

[–1.5, 1.5] by [–1.7, 1.7]

p –2

p2

1 –1

f(x) 5 sin²¹ x

[–2, 2] by [–1, 3.5]

p

–1 1

f(x) 5 cos²¹ x

[–4, 4] by [–2.8, 2.8]

p –2

p 2

f(x) 5 tan²¹ x

Gallery of Functions

CHAPTER 4 Review Exercises

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

The collection of exercises marked in red could be used as a chapter test.

In Exercises 1–8, determine the quadrant of the terminal side of the angle in standard position. Convert degree measures to radians and radian measures to degrees.

1. 5p

2 2. 3p

4

3. -135° 4. -45°

1. 2.

3. 4.

5. 78° 6. 112°

7. p

12 8. 7p

10

In Exercises 9 and 10, determine the angle measure in both degrees and radians. Draw the angle in standard position if its terminal side is obtained as described.

9. A three-quarters counterclockwise rotation 10. Two and one-half counterclockwise rotations

5. 6.

7. 8.

9.

10.

CHAPTER 4 Review Exercises 399

C A

B

α

β 5 cm 12 cm

C A

B a b

c α

β

In Exercises 11–16, the point is on the terminal side of an angle in standard position. Give the smallest positive angle measure in both degrees and radians.

11.

1

2

1

2

In Exercises 17–28, evaluate the expression exactly without a calculator.

17. sin 30° 18. cos 330°

19. tan 1-135°2 20. sec 1-135°2 21. sin 5p

6 22. csc 2p 3 23. sec a- p

3b 24. tan a- 2p 3b 25. csc 270° 26. sec 180°

27. cot 1-90°2 28. tan 360°

In Exercises 29–32, evaluate exactly all six trigonometric functions of the angle. Use reference triangles and not your calculator.

29. - p

6 30. 19p

4

31. -135° 32. 420° 33. Find all six trigonometric functions of a in △ ABC.

11. 12.

13. 14.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33.

47. tan x 6 0 and sin x 7 0 48. sec x6 0 and csc x 7 0

In Exercises 49–52, point P is on the terminal side of angle u.

Evaluate the six trigonometric functions for u.

49. 1-3, 62 50. 112, 72

51. 1-5, -32 52. 14, 92

In Exercises 53–60, use transformations to describe how the graph of the function is related to a basic trigonometric graph. Graph two periods.

53. y = sin 1x + p2 54. y = 3 + 2 cos x 55. y = -cos 1x + p/2) + 4

56. y = -2 - 3 sin 1x - p2

57. y = tan 2x 58. y = -2 cot 3x 59. y = -2 sec x

2 60. y = csc px

In Exercises 61– 66, state the amplitude, period, phase shift, domain, and range for the sinusoid.

61. ƒ1x2 = 2 sin 3x 62. g1x2 = 3 cos 4x 63. ƒ1x2 = 1.5 sin 12x - p/42

64. g1x2 = -2 sin 13x - p/32

65. y = 4 cos 12x - 12 66. g1x2 = -2 cos 13x + 12 In Exercises 67 and 68, graph the function. Then estimate the values of a, b, and h so that ƒ1x2 ≈ a sin 1b1x - h22.

67. ƒ1x2 = 2 sin x - 4 cos x 68. ƒ1x2 = 3 cos 2x - 2 sin 2x

In Exercises 69–72, use a calculator to evaluate the expression.

Express your answer in both degrees and radians.

69. sin^-1 10.7662 70. cos^-1 10.4792 71. tan^-1 1 72. sin^-1 a23

2 b

In Exercises 73–76, use transformations to describe how the graph of the function is related to a basic inverse trigonometric graph. State the domain and range.

73. y = sin^-1 3x 74. y = tan^-1 2x

75. y = sin^-1 13x - 12 + 2 76. y = cos^-1 12x + 12 - 3 In Exercises 77–82, find the exact value of x without using a calculator.

77. sin x = 0.5, p/2 … x … p 78. cos x = 23/2, 0 … x … p

79. tan x = -1, 0 … x … p 80. sec x= 2, p … x … 2p 81. csc x= -1, 0 … x … 2p 82. cot x = - 23, 0 … x … p

In Exercises 83 and 84, describe the end behavior of the function.

83. sin x

x² 84. 3

5 e^-x/12 sin 12x - 32

49. 50.

51. 52.

53. 54.

55.

56.

57. 58.

59. 60.

61. 62.

63.

64.

65. 66.

71. 72.

73. 74.

75. 76.

77.

78.

79.

80.

81.

82.

34. Use a right triangle to determine the values of all trigonomet- ric functions of u, where cos u = 5/7.

35. Use a right triangle to determine the values of all trigonomet- ric functions of u, where tan u = 15/8.

36. Use a calculator in Degree mode to solve cos u = 3/7 if 0° … u … 90°.

37. Use a calculator in Radian mode to solve tan x = 1.35 if p … x … 3p/2.

38. Use a calculator in Radian mode to solve sin x = 0.218 if 0 … x … 2p.

In Exercises 39– 44, solve the right △ ABC.

34.

35.

39. a = 35°, c = 15 40. b= 8, c = 10 41. b = 48°, a = 7 42. a = 28°, c = 8 43. b = 5, c = 7 44. a= 2.5, b = 7.3 In Exercises 45–48, x is an angle in standard position with 0 … x … 2p. Determine the quadrant of x.

45. sin x 6 0 and tan x 7 0 46. cos x 6 0 and csc x 7 0

52ft

18 ft Ground level

Tunnel First hill of

The Beast

P Q

22° 4 mi

L

102. Storing Hay A 75-ft-long conveyor is used at the Lovelady Farm to put hay bales up for winter storage. The conveyor is tilted to an angle of elevation of 22°.

(a) To what height can the hay be moved?

(b) If the conveyor is repositioned to an angle of 27°, to what height can the hay be moved?

103. Swinging Pendulum In the Hardy Boys Adventure While the Clock Ticked, the pendulum of the grandfather clock at the Purdy place is 44 in. long and swings through an arc of 6°. Find the length of the arc that the pendulum traces.

104. Finding Area A windshield wiper arm on a vintage 1994 Plymouth Acclaim is 20 in. long and has a blade 16 in. long.

If the wiper sweeps through an angle of 110°, how large an area does the wiper blade clean? (See Exercise 71 in Section 4.1.)

105. Modeling Low Temperatures The average daily low temperature (°F) in Fairbanks, Alaska, can be modeled by the equation

T1x2 = 31.2 sin c2p

365 1x - 1062 d + 20.4,

where x is time in days, with x = 1 representing January 1.

How many days a year would you expect the low temperature to be above freezing (32°F)?

Source: Weather.com, 2012.

106. Taming The Beast The Beast is a featured roller coaster at the King Island’s amusement park just north of Cincinnati.

On its first and biggest hill, The Beast drops from a height of 52 ft above the ground along a sinusoidal path to a depth 18 ft underground as it enters a frightening tunnel. The mathemati- cal model for this part of track is

h1x2 = 35 cos ax

35b + 17, 0 … x … 110,

where x is the horizontal distance from the top of the hill and h1x2 is the vertical position relative to ground level (both in feet). What is the horizontal distance from the top of the hill to the point where the track reaches ground level?

In Exercises 85–88, evaluate the expression without a calculator.

85. tan 1tan^-1 12 86. cos^-11cos p/32 87. tan 1sin^-1 3/52 88. cos^-1 1cos(-p/7)2

In Exercises 89–92, determine whether the function is periodic. State the period (if applicable), the domain, and the range.

89. ƒ1x2 = ^sec x 90. g1x2 = sin ^x

91. ƒ1x2 = 2x + tan x 92. g1x2 = 2 cos 2x + 3 sin 5x 93. Arc Length Find the length of the arc intercepted by a

central angle of 2p/3 rad in a circle with radius 2.

94. Algebraic Expression Find an algebraic expression equivalent to tan 1cos^-1 x2.

95. Height of Building The angle of elevation of the top of a building from a point 100 m away from the building on level ground is 78°. Find the height of the building.

96. Height of Tree A tree casts a shadow 51 ft long when the angle of elevation of the Sun (measured with the horizon) is 25°. How tall is the tree?

97. Traveling Car From the top of a 150-ft building Flora observes a car moving toward her. If the angle of depression of the car changes from 18° to 42° during the observation, how far does the car travel?

98. Finding Distance A lighthouse L stands 4 mi from the closest point P along a straight shore (see figure). Find the distance from P to a point Q along the shore if ∠PLQ = 22°.

85. 86.

87. 88.

89. 90.

93.

99. Navigation An airplane is flying due east between two signal towers. One tower is due north of the other. The bear- ing from the plane to the north tower is 23°, and to the south tower is 128°. Use a drawing to show the exact location of the plane.

100. Finding Distance The bearings of two points on the shore from a boat are 115° and 123°. Assume the two points are 855 ft apart. How far is the boat from the nearest point on shore if the shore is straight and runs north-south?

101. Height of Tree Dr. Thom Lawson standing on flat ground 62 ft from the base of a Douglas fir measures the an- gle of elevation to the top of the tree as 72°24′. What is the height of the tree?

CHAPTER 4 Review Exercises 401

CHAPTER 4 Project

Modeling the Motion of a Pendulum

As a simple pendulum swings back and forth, its displace- ment can be modeled using a standard sinusoidal equation of the form

y

= a cos 1b1x - h22 + k

where y represents the pendulum’s distance from a fixed point and x represents total elapsed time. In this project, you will use a motion detection device to collect distance and time data for a swinging pendulum, then find a mathematical model that describes the pendulum’s motion.

Collecting the Data

To start, construct a simple pendulum by fastening about 1 m of string to the end of a ball. Set up the Calculator-Based Laboratory (CBL) system with a motion detector or a Calculator-Based Ranger (CBR) system to collect time and distance readings for between 2 sec and 4 sec (enough time to capture at least one complete swing of the pendulum). See the CBL/CBR guidebook for specific setup instructions. Start the pendulum swinging in front of the detector, then activate the system. The data table below shows a sample set of data col- lected as a pendulum swung back and forth in front of a CBR.

Total Elapsed Time (sec)

Distance from the CBR (m)

0 0.665

0.1 0.756

0.2 0.855

0.3 0.903

0.4 0.927

0.5 0.931

0.6 0.897

0.7 0.837

0.8 0.753

0.9 0.663

1.0 0.582

1.1 0.525

1.2 0.509

1.3 0.495

1.4 0.521

1.5 0.575

1.6 0.653

1.7 0.741

1.8 0.825

1.9 0.888

2.0 0.921

Explorations

1.

If you collected motion data using a CBL or CBR, a plot of distance versus time should be shown on your graphing calculator or computer screen. If you don’t have access to a CBL/CBR, enter the data in the sample table into your graphing calculator/computer. Create a scatter plot for the data.

2.

Find values for a, b, h, and k so that the equation y = a cos 1b1x - h22 + k fits the distance versus time data plot. Refer to the information box on page 355 in this chapter to review sinusoidal graph characteristics.

3.

What are the physical meanings of the constants a and k in the modeling equation y = a cos 1b1x - h22 + k?

(Hint: What distances do a and k measure?)

4.

Which, if any, of the values of a, b, h, and/or k would change if you used the equation y = a sin 1b1x - h22 + k to model the data set?

5.

Use your calculator or computer to find a sinusoidal regression equation to model this data set (see your grapher’s guidebook for instructions on how to do this). If your calculator/computer uses a different sinusoidal form, compare it to the modeling equation you found earlier,

y

= a cos 1b1x - h22 + k.