Chapter 4 Trigonometric Functions

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<strong>Trigonometric</strong> <strong>Functions</strong> 38π 38π 70. − + 2π ⋅ 3= − + 6π 9 9 38π 54π 16π =− + = 9 9 9 71. r = 12 inches, θ = 45° Begin by converting 45° to radians, in order to use the formula s = rθ . π radians π 45°= 45°⋅ = radians 180° 4 Now use the formula s = rθ . π s = rθ = 12⋅ = 3π inches ≈ 9.42 inches 4 72. r = 16 inches, θ = 60° Begin by converting 60° to radians, in order to use the formula s = rθ . π radians π 60°= 60°⋅ = radians 180° 3 Now use the formula s = rθ . π 16π s = rθ = 16 ⋅ = inches ≈ 16.76 inches 3 3 73. r = 8feet, θ = 225° Begin by converting 225° to radians, in order to use the formula s = rθ . π radians 5π 225°= 225°⋅ = radians 180° 4 Now use the formula s = rθ . 5π s = rθ = 8⋅ = 10π feet ≈ 31.42 feet 4 74. r = 9 yards, θ = 315° Begin by converting 315° to radians, in order to use the formula s = rθ . π radians 7π 315°= 315°⋅ = radians 180° 4 Now use the formula s = rθ . 7π 63π s = rθ = 9 ⋅ = yards ≈ 49.48 yards 4 4 75. 6 revolutions per second 6 revolutions 2 π radians 12 π radians = ⋅ = 1 second 1 revolutions 1 seconds = 12 π radians per second 77. 78. 79. 80. 81. 4π 2π − and 3 3 7π 5π − and 6 6 3π 5π − and 4 4 π 7π − and 4 4 π 3π − and 2 2 82. −π and π 83. 84. 55 11π ⋅ 2π = 60 6 35 7π ⋅ 2π = 60 6 85. 3 minutes and 40 seconds equals 220 seconds. 220 22π ⋅ 2π = 60 3 86. 4 minutes and 25 seconds equals 265 seconds. 265 53π ⋅ 2π = 60 6 87. First, convert to degrees. 1 1 360° revolution = revolution ⋅ 6 6 1 revolution 1 = ⋅ 360 °= 60 ° 6 Now, convert 60° to radians. π radians 60π 60°= 60°⋅ = radians 180° 180 π = radians 3 Therefore, 1 6 revolution is equivalent to 60° or π 3 radians. 76. 20 revolutions per second 20 revolutions 2 π radians 40 π radians = ⋅ = 1 second 1 revolution 1 second = 40 π radians per second 494 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
PreCalculus 4E Section 4.1 88. First, convert to degrees. 1 1 360 o revolutions = revolutions ⋅ 3 3 1 revolution 1 = ⋅ 360 o = 120 o 3 Now, convert 120° to radians. π radians 120π 120°= 120°⋅ = radians 180° 180 2π = radians 3 Therefore, 1 3 revolution is equivalent to 120° or 2 π 3 radians. 89. The distance that the tip of the minute hand moves is given by its arc length, s. Since s = rθ , we begin by finding r and θ . We are given that r = 8 inches. The minute hand moves from 12 to 2 o'clock, or 1 6 of a complete revolution. The formula s = rθ can only be used when θ is expressed in radians. We must convert 1 revolution to radians. 6 1 1 2 π radians revolution = revolution ⋅ 6 6 1 revolution π = radians 3 The distance the tip of the minute hand moves is ⎛π ⎞ 8π s = rθ = (8 inches) ⎜ ⎟= inches ⎝ 3 ⎠ 3 ≈ 8.38 inches. 90. The distance that the tip of the minute hand moves is given by its arc length, s. Since s = rθ , we begin by finding r and θ . We are given that r = 6 inches. The minute hand moves from 12 to 4 o’clock, or 1 of a complete revolution. The formula 3 s = rθ can only be used when θ is expressed in radians. We must convert 1 revolution to radians. 3 1 1 2 π radians revolution = revolution ⋅ 3 3 1 revolution 2π = radians 3 The distance the tip of the minute hand moves is ⎛2π ⎞ 12π s = rθ = (6 inches) ⎜ ⎟= inches ⎝ 3 ⎠ 3 = 4π inches ≈12.57 inches. 91. The length of each arc is given by s = rθ . We are given that r = 24 inches and θ = 90 ° . The formula s = rθ can only be used when θ is expressed in radians. π radians 90π 90°= 90°⋅ = radians 180° 180 π = radians 2 The length of each arc is ⎛π ⎞ s = rθ = (24 inches) ⎜ ⎟= 12π inches ⎝ 2 ⎠ ≈ 37.70 inches. 92. The distance that the wheel moves is given by s = rθ . We are given that r = 80 centimeters and θ = 60°. The formula s = rθ can only be used when θ is expressed in radians. π radians 60π 60°= 60°⋅ = radians 180° 180 π = radians 3 The length that the wheel moves is ⎛π ⎞ 80π s = rθ = (80 centimeters) ⎜ ⎟ = centimeters ⎝ 3⎠ 3 ≈ 83.78 centimeters. s 93. Recall that θ = . We are given that r s = 8000 miles and r = 4000 miles. s 8000 miles θ = = = 2 radians r 4000 miles Now, convert 2 radians to degrees. 180 o 2 radians = 2 radians ⋅ ≈ 114.59 o π radians s 94. Recall that θ = . We are given that r s = 10,000 miles and r = 4000 miles. s 10,000 miles θ = = = 2.5 radians r 4000 miles Now, convert 2.5 radians to degrees. 180 o 2.5 radians⋅ ≈ 143.24 o 2 π radians 495 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Page 3 and 4: PreCalculus 4E Section 4.1 16. 17.
Page 5: PreCalculus 4E Section 4.1 48. 55.
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Page 15 and 16: PreCalculus 4E Section 4.2 0 π 57.
Page 17 and 18: PreCalculus 4E Section 4.2 π H = 1
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Page 23 and 24: PreCalculus 4E Section 4.3 π π π
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Chapter 4 Trigonometric Functions

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