Textbook Question
The propeller of a 90-horsepower outboard motor at full throttle rotates at exactly 5000 revolutions per min. Find the angular speed of the propeller in radians per second.
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 3.55
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
cot s = 0.5022
Verified step by step guidance1
Recognize that \( \cot(s) = \frac{1}{\tan(s)} \). Therefore, \( \tan(s) = \frac{1}{0.5022} \).
Calculate \( \tan(s) \) using the reciprocal of 0.5022.
Use a calculator to find the angle \( s \) in radians for which \( \tan(s) \) equals the calculated value.
Ensure that the angle \( s \) is within the interval \([0, \frac{\pi}{2}]\).
Round the value of \( s \) to four decimal places.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cotangent Function
The cotangent function, denoted as cot(s), is the reciprocal of the tangent function. It is defined as cot(s) = cos(s)/sin(s). In the context of the unit circle, cotangent represents the ratio of the adjacent side to the opposite side in a right triangle. Understanding cotangent is essential for solving equations involving angles and their trigonometric ratios.
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Introduction to Cotangent Graph
Inverse Trigonometric Functions
Inverse trigonometric functions, such as arccot or cot^(-1), are used to find the angle that corresponds to a given trigonometric ratio. For example, if cot(s) = 0.5022, we can use the inverse cotangent function to determine the angle s. These functions are crucial for solving equations where the angle is unknown and must be derived from a known ratio.
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4:28Introduction to Inverse Trig Functions
Interval [0, π/2]
The interval [0, π/2] represents the range of angles from 0 to 90 degrees, where all trigonometric functions are positive. This interval is significant when solving trigonometric equations because it restricts the possible values of s, ensuring that the solution is within the first quadrant. Understanding the implications of this interval helps in determining the correct angle that satisfies the given cotangent value.
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1:48Example 2
Related Practice
Textbook Question
Find the angular speed ω for each of the following.
a wind turbine with blades turning at a rate of 15 revolutions per minute
Textbook Question
Find the exact value of s in the given interval that has the given circular function value.
[ π , 3π/2] ; sec s = ―2√3/3
Textbook Question
Work each problem.
Consider each angle in standard position having the given radian measure. In what quadrant does the terminal side lie?
4
Textbook Question
Work each problem.
Consider each angle in standard position having the given radian measure. In what quadrant does the terminal side lie?
-2
Textbook Question
Work each problem.
Consider each angle in standard position having the given radian measure. In what quadrant does the terminal side lie?
7
2
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