Textbook Question
Convert each radian measure to degrees. Write answers to the nearest minute. See Example 2(c).
3.06
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 61
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
tan s = 0.2126
Verified step by step guidance1
Identify the given equation: \(\tan s = 0.2126\), where \(s\) is in the interval \([0, \frac{\pi}{2}]\).
Recall that to find \(s\), you need to use the inverse tangent function (also called arctangent), which is written as \(\arctan\) or \(\tan^{-1}\).
Apply the inverse tangent to both sides of the equation to isolate \(s\): \(s = \arctan(0.2126)\).
Use a calculator set to radians mode to find the approximate value of \(s\) by evaluating \(\arctan(0.2126)\).
Round the result to four decimal places to get the final 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.
Understanding the Tangent Function
The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic and defined for all angles except where cosine is zero. Knowing how to interpret tan s = 0.2126 helps in finding the angle s whose tangent value matches this ratio.
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Introduction to Tangent Graph
Inverse Tangent (Arctan) Function
The inverse tangent function, denoted arctan or tan⁻¹, is used to find the angle whose tangent is a given number. Since tan s = 0.2126, s = arctan(0.2126) gives the angle in radians. This function returns values typically in the interval (-π/2, π/2), which includes the given domain [0, π/2].
Recommended video:
3:17Inverse Tangent
Radian Measure and Interval Constraints
Angles can be measured in radians, where π radians equal 180 degrees. The problem restricts s to the interval [0, π/2], meaning s is between 0 and 90 degrees. Understanding this interval ensures the correct solution is chosen, especially since tangent is positive and increasing in this 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).
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Textbook Question
Work each problem. See Example 5. Irrigation Area A center-pivot irrigation system provides water to a sector-shaped field as shown in the figure. Find the area of the field if θ = 40.0° and r = 152 yd.
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Textbook Question
Convert each radian measure to degrees. Write answers to the nearest minute. See Example 2(c).
0.3417
Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
tan 6.29
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?