Textbook Question
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
sin s = 0.9918
Ch. 3 - Radian Measure and The Unit Circle
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 4, Problem 65
Find the approximate value of s, to four decimal places, in the interval [0 , π/2] that makes each statement true.
sec s = 1.0806
Verified step by step guidance1
Recall the definition of the secant function: \(\sec s = \frac{1}{\cos s}\). This means that \(\cos s = \frac{1}{\sec s}\).
Substitute the given value of \(\sec s\) into the equation: \(\cos s = \frac{1}{1.0806}\).
Calculate the value of \(\cos s\) from the above expression (you can do this with a calculator, but do not finalize the answer here).
Use the inverse cosine function to find \(s\): \(s = \arccos(\cos s)\), where \(s\) is in the interval \([0, \frac{\pi}{2}]\).
Express the value of \(s\) in radians and round it to four decimal places as required.
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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 the angle.
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Graphs of Secant and Cosecant Functions
Inverse Trigonometric Functions
Inverse trigonometric functions, such as arccos, are used to find the angle corresponding to a given trigonometric value. After finding cos(s) from sec(s), applying arccos helps determine the angle s within the specified interval [0, π/2].
Recommended video:
4:28Introduction to Inverse Trig Functions
Domain and Range Restrictions
The problem restricts s to the interval [0, π/2], which corresponds to the first quadrant where cosine values are positive. This restriction ensures the solution is unique and helps in selecting the correct angle from the inverse cosine function.
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4:22Domain and Range of Function Transformations
Related Practice
Textbook Question
Convert each radian measure to degrees. Write answers to the nearest minute. See Example 2(c).
-5.01095
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Textbook Question
Find each exact function value. See Example 3.
sin π/3
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Textbook Question
Find the exact value of s in the given interval that has the given circular function value.
[π/2, π] ; sin s = 1/2
Textbook Question
Find the exact value of s in the given interval that has the given circular function value.
[π, 3π/2] ; tan s = √3
Textbook Question
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
cos s = 0.7826