# Trigonometric Functions

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1 CHAPTER Angles and Their Measures 4.2 Trigonometric Functions of Acute Angles 4.3 Trigonometry Extended: The Circular Functions 4.4 Graphs of Sine and Cosine: Sinusoids 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant Trigonometric Functions When the motion of an object causes air molecules to vibrate, we hear a sound. We measure sound according to its pitch and loudness, which are attributes associated with the frequency and amplitude of sound waves. As we shall see, it is the branch of mathematics called trigonometry that enables us to analyze waves of all kinds; indeed, that is only one application of this powerful analytical tool. See page 393 for an application of trigonometry to sound waves. 4.6 Graphs of Composite Trigonometric Functions 4.7 Inverse Trigonometric Functions 4.8 Solving Problems with Trigonometry 319

6 324 CHAPTER 4 Trigonometric Functions A nautical mile (naut mi) is the length of 1 minute of arc along Earth s equator. Figure 4.6 shows, though not to scale, a central angle AOB of Earth that measures 1/60 of a degree. It intercepts an arc 1 naut mi long. The arc length formula allows us to convert between nautical miles and statute miles (stat mi), the familiar land mile of 5280 ft. B Nautical mile A O FIGURE 4.6 Although Earth is not a perfect sphere, its diameter is, on average, stat mi. A nautical mile is 1 of Earth s circumference at the equator. EXAMPLE 6 Converting to Nautical Miles Megan McCarty, a pilot for Western Airlines, frequently pilots flights from Boston to San Francisco, a distance of 2698 stat mi. Captain McCarty s calculations of flight time are based on nautical miles. How many nautical miles is it from Boston to San Francisco? SOLUTION The radius of the Earth at the equator is approximately 3956 stat mi. Convert 1 minute to radians: 1 = a 1 60 b * p rad 180 = p 10,800 rad. Now we can apply the formula s = ru: p 1 naut mi = a b stat mi 10, stat mi 1 stat mi = a 10,800 b naut mi 3956p 0.87 naut mi The distance from Boston to San Francisco is 2698 stat mi = 2698 # 10, naut mi. 3956p Now try Exercise 51. Distance Conversions 1 stat mi 0.87 naut mi 1 naut mi 1.15 stat mi

10 328 CHAPTER 4 Trigonometric Functions In Exercises 63 66, find the difference in longitude between the given cities. 63. Atlanta and San Francisco 64. New York and San Diego 65. Minneapolis and Chicago 66. Miami and Seattle In Exercises 67 70, assume that the two cities have the same longitude (that is, assume that one is directly north of the other), and find the distance between them in nautical miles. 67. San Diego and Los Angeles 68. Seattle and San Francisco 69. New Orleans and Minneapolis 70. Detroit and Atlanta 71. Group Activity Area of a Sector A sector of a circle (shaded in the figure) is a region bounded by a central angle of a circle r and its intercepted arc. Use the fact that θ the areas of sectors are proportional to r their central angles to prove that Extending the Ideas 72. Area of a Sector Use the formula A = (1/2)r 2 u to determine the area of the sector with given radius r and central angle u. (a) r = 5.9 ft, u = p/5 (b) r = 1.6 km, u = Navigation Control tower A is 60 mi east of control tower B. At a certain time an airplane is on bearings of 340 from tower A and 37 from tower B. Use a drawing to model the exact location of the airplane. 74. Bicycle Racing Ben Scheltz s bike wheels are 28 in. in diameter, and for high gear the pedal sprocket is 9 in. in diameter and the wheel sprocket is 3 in. in diameter. Find the angular speed in radians per second of the wheel and of both sprockets when Ben reaches his peak racing speed of 66 ft/sec in high gear. 28 in. A = 1 2 r2 u where r is the radius and u is in radians.

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