Textbook Question
Convert each radian measure to degrees. Write answers to the nearest minute. See Example 2(c).
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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 63
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
Verified step by step guidance1
Identify the problem: We need to find the value of \( s \) in the interval \( [0, \frac{\pi}{2}] \) such that \( \sin s = 0.9918 \).
Recall that to find an angle \( s \) given \( \sin s = a \), we use the inverse sine function (also called arcsine), denoted as \( \sin^{-1} \) or \( \arcsin \). So, \( s = \arcsin(0.9918) \).
Make sure the value \( 0.9918 \) is within the valid range for sine, which is \( [-1, 1] \). Since it is, the inverse sine function is defined.
Calculate \( s = \arcsin(0.9918) \) using a calculator set to radians mode to get the angle in radians.
Round the result to four decimal places to get the approximate value of \( s \) within the interval \( [0, \frac{\pi}{2}] \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sine Function and Its Range
The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. Its values range between -1 and 1, and within the interval [0, π/2], sine values increase from 0 to 1. Understanding this helps identify valid solutions for the equation sin s = 0.9918.
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Domain and Range of Function Transformations
Inverse Sine Function (Arcsin)
The inverse sine function, denoted as arcsin or sin⁻¹, returns the angle whose sine is a given value. It is used to find the angle s when sin s is known. Since arcsin outputs values in [-π/2, π/2], restricting the domain to [0, π/2] ensures a unique solution for s.
Recommended video:
4:03Inverse Sine
Radian Measure and Interval Constraints
Angles in trigonometry are often measured in radians, where π radians equal 180 degrees. The interval [0, π/2] corresponds to angles from 0 to 90 degrees, representing the first quadrant where sine values are positive and increasing. This constraint limits the solution to a specific range.
Recommended video:
5:04Converting between Degrees & Radians
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 the approximate value of s, to four decimal places, in the interval [0 , π/2] that makes each statement true.
sec s = 1.0806
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Textbook Question
Find each exact function value. See Example 3.
sin π/3
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Textbook Question
Work each problem. See Example 5. Arc Length A circular sector has an area of 50 in² . The radius of the circle is 5 in. What is the arc length of the sector?
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