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 = 4.82 m , θ = 60°
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 15
Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.
s = ―π
Verified step by step guidance1
Identify the given angle: here, the angle is \(s = -\pi\) radians. This means the angle is negative, indicating a clockwise rotation of \(\pi\) radians from the positive x-axis.
Recall the unit circle values for \(\pi\) radians: on the unit circle, an angle of \(\pi\) radians corresponds to the point \((-1, 0)\), where the x-coordinate is \(\cos \pi\) and the y-coordinate is \(\sin \pi\).
Use the even-odd properties of sine and cosine to find values for negative angles: since \(\sin(-\theta) = -\sin \theta\) and \(\cos(-\theta) = \cos \theta\), apply these to \(s = -\pi\).
Calculate \(\sin s\) using the property: \(\sin(-\pi) = -\sin \pi\). Since \(\sin \pi = 0\), this gives \(\sin(-\pi) = 0\).
Calculate \(\cos s\) and \(\tan s\): \(\cos(-\pi) = \cos \pi = -1\), and \(\tan s = \frac{\sin s}{\cos s} = \frac{0}{-1} = 0\).
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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. Understanding how to locate angles like -π on the unit circle is essential for finding exact trigonometric values.
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Introduction to the Unit Circle
Sine, Cosine, and Tangent Functions
Sine and cosine functions represent the y and x coordinates, respectively, of a point on the unit circle corresponding to an angle. Tangent is the ratio of sine to cosine (tan s = sin s / cos s). Knowing these definitions helps in calculating exact values for any real number angle, including negative angles like -π.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Properties of Trigonometric Functions for Negative Angles
Trigonometric functions have specific symmetry properties: sine is an odd function (sin(-s) = -sin s), cosine is even (cos(-s) = cos s), and tangent is odd (tan(-s) = -tan s). These properties allow simplification when evaluating trigonometric values at negative angles such as -π.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). ―300°
Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 150°
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 = 1.38 ft , θ = 5π/6 radians
Textbook Question
Use the formula ω = θ/t to find the value of the missing variable.
θ = 3.871 radians, t = 21.47 sec
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Textbook Question
Convert each radian measure to degrees.
-11π/18
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