Textbook Question
Use the formula ω = θ/t to find the value of the missing variable.
ω = 0.91 radian per min, t = 8.1 min
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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 13
Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.
s = 2π
Verified step by step guidance1
Recall that the sine, cosine, and tangent functions are periodic with period \(2\pi\). This means for any real number \(s\), \(\sin(s + 2\pi) = \sin s\), \(\cos(s + 2\pi) = \cos s\), and \(\tan(s + 2\pi) = \tan s\).
Since \(s = 2\pi\), recognize that this corresponds to one full rotation around the unit circle, bringing the angle back to the starting point.
Evaluate \(\sin(2\pi)\) by considering the \(y\)-coordinate of the point on the unit circle at angle \(2\pi\). The point is \((\cos 2\pi, \sin 2\pi)\).
Evaluate \(\cos(2\pi)\) by considering the \(x\)-coordinate of the point on the unit circle at angle \(2\pi\).
Find \(\tan(2\pi)\) using the definition \(\tan s = \frac{\sin s}{\cos s}\), substituting the values found for \(\sin(2\pi)\) and \(\cos(2\pi)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Angle Measurement
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on the unit circle, where the x-coordinate gives the cosine value and the y-coordinate gives the sine value of the angle.
Recommended video:
06:11
Introduction to the Unit Circle
Periodic Properties of Trigonometric Functions
Sine, cosine, and tangent functions are periodic, meaning their values repeat at regular intervals. For sine and cosine, the period is 2π, so sin(s + 2π) = sin s and cos(s + 2π) = cos s, which helps find exact values for angles like s = 2π.
Recommended video:
5:33Period of Sine and Cosine Functions
Definition of Tangent in Terms of Sine and Cosine
Tangent of an angle s is defined as tan s = sin s / cos s, provided cos s ≠ 0. Understanding this relationship allows calculation of tangent values once sine and cosine are known, especially for standard angles on the unit circle.
Recommended video:
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Related Practice
Textbook Question
Find the length to three significant digits of each arc intercepted by a central angle in a circle of radius r. See Example 1. r = 12.3 cm , θ = 2π/3 radians
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Textbook Question
Convert each radian measure to degrees.
8π/3
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Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 30°
609
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Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 90°
620
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Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 60°
648
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