Textbook Question
Find each exact function value. See Example 3.
sin 5π/6
7
views
Ch. 3 - Radian Measure and The Unit Circle
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
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 75
Find the exact values of s in the given interval that satisfy the given condition.
[0 , 2π) ; cos² s = 1/2
Verified step by step guidance1
Start with the given equation: \(\cos^{2} s = \frac{1}{2}\).
Take the square root of both sides to solve for \(\cos s\): \(\cos s = \pm \sqrt{\frac{1}{2}}\).
Simplify the square root: \(\cos s = \pm \frac{\sqrt{2}}{2}\).
Recall the unit circle values where \(\cos s = \pm \frac{\sqrt{2}}{2}\), which correspond to angles \(s = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) within the interval \([0, 2\pi)\).
List these values as the exact solutions for \(s\) in the given interval.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions and Their Squares
The cosine function, cos(s), measures the horizontal coordinate of a point on the unit circle at angle s. Squaring the cosine, cos²(s), means taking the square of this value, which is always non-negative and ranges from 0 to 1. Understanding how cos²(s) relates to cos(s) is essential for solving equations involving squared trigonometric functions.
Recommended video:
6:04
Introduction to Trigonometric Functions
Unit Circle and Angle Intervals
The unit circle represents all angles s from 0 to 2π radians, corresponding to one full rotation. Knowing how cosine values correspond to points on the unit circle helps identify all angles s within the interval [0, 2π) that satisfy the equation. This includes recognizing symmetry and periodicity of cosine.
Recommended video:
06:11Introduction to the Unit Circle
Solving Trigonometric Equations
To solve equations like cos²(s) = 1/2, one must first find cos(s) = ±√(1/2) = ±√2/2. Then, determine all angles s in the given interval where cosine equals these values. This involves using inverse cosine functions and considering all solutions within the specified domain.
Recommended video:
4:34How to Solve Linear Trigonometric Equations
Related Practice
Textbook Question
Find each exact function value. See Example 3.
tan 5π/3
2
views
Textbook Question
Find the exact values of s in the given interval that satisfy the given condition.
[0, 2π) ; sin s = -√3 / 2
Textbook Question
Find each exact function value. See Example 3.
sin π/2
5
views
Textbook Question
Find the exact values of s in the given interval that satisfy the given condition.
[-2π , π) ; 3 tan² s = 1
Textbook Question
Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.
s = 2.5
<IMAGE>