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 2.3.14
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'
Verified step by step guidance1
Convert the angle from degrees and minutes to decimal degrees. Since 1 minute is \( \frac{1}{60} \) of a degree, calculate \( 45' = \frac{45}{60} = 0.75 \) degrees. Then, add this to 145 degrees to get the total angle: \( 145 + 0.75 = 145.75^\circ \).
Recall that \( \csc \theta = \frac{1}{\sin \theta} \). So, to find \( \csc 145^\circ 45' \), you first need to find \( \sin 145.75^\circ \).
Use a calculator set to degree mode to find \( \sin 145.75^\circ \).
Calculate the reciprocal of the sine value to find the cosecant: \( \csc 145.75^\circ = \frac{1}{\sin 145.75^\circ} \).
Round your final answer 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.
Understanding the Cosecant Function
The cosecant function, csc(θ), is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). To find csc of an angle, first find the sine of that angle, then take its reciprocal. This is essential for evaluating expressions involving csc.
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Converting Degrees and Minutes to Decimal Degrees
Angles given in degrees and minutes must be converted to decimal degrees before using a calculator. Since 1 minute equals 1/60 of a degree, convert by adding minutes divided by 60 to the degrees. For example, 145° 45' = 145 + 45/60 = 145.75°.
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Using a Calculator to Approximate Trigonometric Values
After simplifying and converting the angle, use a scientific calculator set to degree mode to find the sine value. Then compute the reciprocal for cosecant. Round the final answer to six decimal places as required for precision.
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Related Practice
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Find exact values of the six trigonometric functions for each angle A.
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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.
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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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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 70° = 2 cos² 35° - 1
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