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Problem 80
Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. cot(θ + 180°)
Problem 81
Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. sin(-θ)
Problem 85
Concept Check Work each problem. Without using a calculator, determine which of the following numbers is closest to sin 115°: -0.9, -0.1, 0, 0.1, or 0.9.
Problem 87
Concept Check Work each problem. For what angles θ between 0° and 360° is cos θ = sin θ true?
Problem 1
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I: 1. sin 83°
Column II: A. 88.09084757°; B. 63.25631605°; C. 1.909152433°; D. 17.45760312°; E. 0.2867453858; F. 1.962610506; G. 14.47751219°; H. 1.015426612; I. 1.051462224; J. 0.9925461516
Problem 2.3.10
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I: 1.
cot⁻¹ 30
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516
Problem 2
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I:
cos⁻¹ 0.45
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516
Problem 2.3.72
(Modeling) Grade Resistance Solve each problem. See Example 3. A 3000-lb car traveling uphill has a grade resistance of 150 lb. Find the angle of the grade to the nearest tenth of a degree.
Problem 2.3.25
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
cot(90°-4.72°)
Problem 2.3.9
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I: 1.
csc⁻¹ 4
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516
Problem 2.3.83
(Modeling) Fish's View of the World The figure shows a fish's view of the world above the surface of the water. (Data from Walker, J., 'The Amateur Scientist,' Scientific American.) Suppose that a light ray comes from the horizon, enters the water, and strikes the fish's eye. Assume that this ray gives a value of 90° for angle θ₁ in the formula for Snell's law. (In a practical situation, this angle would probably be a little less than 90°.) The speed of light in water is about 2.254 x 10⁸ m per sec. Find angle θ₂ to the nearest tenth.
Problem 2.3.91
(Modeling) Length of a Sag Curve When a highway goes downhill and then uphill, it has a sag curve. Sag curves are designed so that at night, headlights shine sufficiently far down the road to allow a safe stopping distance. See the figure. S and L are in feet. The minimum length L of a sag curve is determined by the height h of the car's headlights above the pavement, the downhill grade θ₁ < 0°, the uphill grade θ₂ > 0°, and the safe stopping distance S for a given speed limit. In addition, L is dependent on the vertical alignment of the headlights. Headlights are usually pointed upward at a slight angle α above the horizontal of the car. Using these quantities, for a 55 mph speed limit, L can be modeled by the formula (θ₂ - θ₁)S² L = ————————— , 200(h + S tan α) where S < L. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) Compute length L, to the nearest foot, if h = 1.9 ft, α = 0.9°, θ₁ = -3°, θ₂ = 4°, and S = 336 ft.
Problem 2.3.70
(Modeling) Grade Resistance Solve each problem. See Example 3. Find the grade resistance, to the nearest ten pounds, for a 2400-lb car traveling on a -2.4° downhill grade.
Problem 2.3.38
Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
cos θ = 0.85536428
Problem 2.3.51
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. sin 10° + sin 10° = sin 20°
Problem 2.3.45
Use a calculator to evaluate each expression. sin 35° cos 55° + cos 35° sin 55°
Problem 2.3.30
Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
tan θ = 6.4358841
Problem 2.3.56
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. 2 cos 38°22' = cos 76°44'
Problem 2.3.74
(Modeling) Grade Resistance Solve each problem. See Example 3. A car traveling on a -3° downhill grade has a grade resistance of -145 lb. Determine the weight of the car to the nearest hundred pounds.
Problem 2.3.26
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
cos(90°-3.69°)
Problem 2.3.58
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. ½ sin 40° = sin [½ (40°)]
Problem 2.3.12
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
cos 41° 24'
Problem 2.3.52
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. cos 40° = 2 cos 20°
Problem 2.3.62
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. cos(30° + 20°) = cos 30° + cos 20°
Problem 2.3.34
Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
csc θ = 1.3861147
Problem 2.3.40
Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2. cot θ = 0.21563481
Problem 2.3.27
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
1/csc(90°-51°)
Problem 2.3.66
Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. cos θ = 0.10452846
Problem 2.3.18
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1. tan(-80° 06')
Problem 2.3.36
Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
sec θ = 1.1606249