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 3.57
Find the exact value of s in the given interval that has the given circular function value.
[ 0, π/2] ; cos s = √2/2
Verified step by step guidance1
Identify the range of the angle: The problem specifies the interval \([0, \pi/2]\), which means we are looking for an angle in the first quadrant.
Recall the cosine values for special angles: In the first quadrant, \(\cos(\pi/4) = \frac{\sqrt{2}}{2}\).
Compare the given cosine value with known values: The given value \(\cos s = \frac{\sqrt{2}}{2}\) matches the cosine of \(\pi/4\).
Verify the angle is within the specified interval: Since \(\pi/4\) is within \([0, \pi/2]\), it is a valid solution.
Conclude that the exact value of \(s\) is \(\pi/4\) within the given interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine Function
The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all real numbers and is periodic with a period of 2π. The cosine function takes values between -1 and 1, and specific angles yield well-known cosine values, such as cos(π/4) = √2/2.
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Graph of Sine and Cosine Function
Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a crucial tool in trigonometry, as it allows for the visualization of the sine and cosine functions. The coordinates of any point on the unit circle correspond to the cosine and sine of the angle formed with the positive x-axis, making it easier to determine exact values for trigonometric functions.
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06:11Introduction to the Unit Circle
Principal Values of Trigonometric Functions
Principal values refer to the specific angles within a defined interval where a trigonometric function takes a particular value. For cosine, the principal values are typically found in the intervals [0, π] for angles in radians. In this case, since we are looking for s in the interval [0, π/2], we need to identify the angle whose cosine equals √2/2, which is π/4.
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6:04Introduction to Trigonometric Functions
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
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Find a calculator approximation to four decimal places for each circular function value.
sec 7.3159
Textbook Question
Find the exact value of s in the given interval that has the given circular function value.
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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
Give an expression that generates all angles coterminal with an angle of π/6 radian. Let n represent any integer.
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