Textbook Question
CONCEPT PREVIEW Find the area of each sector.
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 11
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/2
Verified step by step guidance1
Identify the angle given: here, the angle \( s = \frac{\pi}{2} \) radians, which corresponds to 90 degrees on the unit circle.
Recall the definitions of sine, cosine, and tangent on the unit circle: for an angle \( s \), \( \sin s \) is the y-coordinate, \( \cos s \) is the x-coordinate, and \( \tan s = \frac{\sin s}{\cos s} \) provided \( \cos s \neq 0 \).
Locate the point on the unit circle corresponding to \( s = \frac{\pi}{2} \). This point is at the top of the circle, where the coordinates are \( (0, 1) \).
From the coordinates, determine \( \sin s = 1 \) and \( \cos s = 0 \).
Calculate \( \tan s = \frac{\sin s}{\cos s} = \frac{1}{0} \), which is undefined because division by zero is not possible.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Radian Measure
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to arc lengths on this circle, where π radians equal 180 degrees. Understanding the position of an angle like π/2 on the unit circle helps determine the sine, cosine, and tangent values.
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Introduction to the Unit Circle
Definition of Sine, Cosine, and Tangent
Sine of an angle is the y-coordinate, cosine is the x-coordinate of the corresponding point on the unit circle. Tangent is the ratio of sine to cosine (tan s = sin s / cos s). These definitions allow exact evaluation of trigonometric functions for special angles such as π/2.
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Exact Values of Trigonometric Functions at Special Angles
Certain angles like 0, π/6, π/4, π/3, and π/2 have well-known exact sine, cosine, and tangent values. For s = π/2, sin s = 1, cos s = 0, and tan s is undefined due to division by zero. Memorizing or deriving these values is essential for solving trigonometric problems precisely.
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Related Practice
Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π.
800°
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Textbook Question
Use the formula ω = θ/t to find the value of the missing variable.
ω = 2π/3 radians per sec, t = 3 sec
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Textbook Question
Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.
Textbook Question
Convert each radian measure to degrees.
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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°
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