Textbook Question
CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II.
I. N 70° W
II. 1. A. B. C. 2. 3. 4. D. E. F. 5. N 70° W 6. 7. G. H. 8. 9. 10. I. J.
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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 2.3.36
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
Verified step by step guidance1
Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\cos \theta = \frac{1}{\sec \theta}\).
Substitute the given value of \(\sec \theta\) into the equation: \(\cos \theta = \frac{1}{1.1606249}\).
Calculate the value of \(\cos \theta\) using the reciprocal of the given secant value (do not compute the final number here, just set up the expression).
Use the inverse cosine function to find \(\theta\): \(\theta = \cos^{-1}(\text{value from step 3})\). This will give \(\theta\) in radians or degrees depending on your calculator settings.
Since the problem asks for \(\theta\) in degrees within the interval \([0^\circ, 90^\circ)\), ensure your calculator is set to degrees and that the angle you find lies within this interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Secant Function
The secant function, sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Understanding this relationship allows you to convert the given secant value into a cosine value, which is often easier to work with when solving for θ.
Recommended video:
6:22
Graphs of Secant and Cosecant Functions
Inverse Trigonometric Functions
To find the angle θ from a trigonometric value, you use inverse functions such as arccos or arcsec. Since sec θ = 1.1606249, you first find cos θ = 1/1.1606249, then apply the inverse cosine function to determine θ within the specified interval.
Recommended video:
4:28Introduction to Inverse Trig Functions
Domain and Range Restrictions
The problem restricts θ to the interval [0°, 90°), which corresponds to the first quadrant where cosine values are positive. This restriction ensures a unique solution for θ and guides the selection of the correct angle from the inverse function's output.
Recommended video:
4:22Domain and Range of Function Transformations
Related Practice
Textbook Question
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
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Textbook Question
CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II.
I. 10. N 70° E
II. 1. A. B. C. 2. 3. 4. D. E. F. 5. 6. 7. G. H. 8. 9. I. J.
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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. ½ sin 40° = sin [½ (40°)]
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Textbook Question
Solve each problem. See Examples 1 and 2. Flying Distance The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph takes 1.8 hr to go from A to B. Find the distance from B to C.
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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. tan(-80° 06')
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