Textbook Question
(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.
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Ch. 2 - Acute Angles and Right Triangles
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 3, Problem 2.3.16
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 421° 30'
Verified step by step guidance1
Convert the angle from degrees and minutes to a decimal degree format. Since 1 minute is \( \frac{1}{60} \) of a degree, calculate the decimal degrees as \( 421 + \frac{30}{60} \).
Simplify the angle by reducing it within the standard range of \(0^\circ\) to \(360^\circ\) using the formula \( \theta_{reduced} = \theta - 360k \), where \(k\) is an integer chosen so that \( \theta_{reduced} \) lies in the desired range.
Use the tangent function on the simplified angle: \( \tan(\theta_{reduced}) \).
Use a calculator to find the approximate value of \( \tan(\theta_{reduced}) \), making sure the calculator is set to degree mode.
Round the result to six decimal places as requested.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angle Conversion Between Degrees and Minutes
Angles given in degrees and minutes must be converted into decimal degrees before calculation. Since 1 minute equals 1/60 of a degree, 30' is converted to 0.5°, making 421° 30' equal to 421.5°. This conversion is essential for accurate input into calculators.
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Converting between Degrees & Radians
Coterminal Angles and Angle Reduction
Angles larger than 360° can be reduced by subtracting multiples of 360° to find a coterminal angle within the standard 0° to 360° range. This simplification helps in evaluating trigonometric functions more easily and avoids calculator errors.
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04:46Coterminal Angles
Using a Calculator to Find Trigonometric Values
After simplifying the angle, use a scientific calculator set to degree mode to find the tangent value. Ensure the calculator is in the correct mode and round the result to six decimal places as required for precision.
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Related Practice
Textbook Question
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°
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Textbook Question
Find exact values of the six trigonometric functions for each angle A.
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Textbook Question
CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II.
I. 8. 270°
II.
1. A. B. C. 2. 3. 4. D. E. F. 5. 6. 7. G. H. 9. 10. I. J.
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Textbook Question
(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.
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Textbook Question
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. csc 145° 45'
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