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 3.59
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
Verified step by step guidance1
Recognize that the secant function, \( \sec(s) \), is the reciprocal of the cosine function, \( \cos(s) \). Therefore, \( \sec(s) = -\frac{2\sqrt{3}}{3} \) implies \( \cos(s) = -\frac{3}{2\sqrt{3}} \).
Simplify \( \cos(s) = -\frac{3}{2\sqrt{3}} \) by rationalizing the denominator to get \( \cos(s) = -\frac{\sqrt{3}}{2} \).
Identify the reference angle where \( \cos(\theta) = \frac{\sqrt{3}}{2} \). This angle is \( \theta = \frac{\pi}{6} \).
Since \( s \) is in the interval \([\pi, \frac{3\pi}{2}]\), and \( \cos(s) \) is negative, \( s \) must be in the third quadrant.
Determine the angle in the third quadrant by using the reference angle: \( s = \pi + \frac{\pi}{6} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Secant Function
The secant function, denoted as sec(s), is the reciprocal of the cosine function. It is defined as sec(s) = 1/cos(s). Understanding the secant function is crucial for solving problems involving circular functions, as it helps to determine the angle s when given a specific secant value.
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Graphs of Secant and Cosecant Functions
Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the trigonometric functions. The angles and their corresponding sine, cosine, and secant values can be easily visualized on the unit circle, aiding in finding exact values for trigonometric equations.
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06:11Introduction to the Unit Circle
Quadrants and Angle Ranges
Trigonometric functions have different signs in different quadrants of the unit circle. The interval [π, 3π/2] corresponds to the third quadrant, where both sine and cosine are negative. Recognizing the quadrant is essential for determining the correct angle that satisfies the given secant value, as it influences the sign and value of the trigonometric functions.
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6:36Quadratic Formula
Related Practice
Textbook Question
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
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 linear speed v for each of the following.
the tip of the minute hand of a clock, if the hand is 7 cm long
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Textbook Question
Work each problem.
Consider each angle in standard position having the given radian measure. In what quadrant does the terminal side lie?
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Textbook Question
Find the exact value of s in the given interval that has the given circular function value.
[ 0, π/2] ; cos s = √2/2