Textbook Question
Find each exact function value. See Example 2.
sin (-4π/3)
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 29
The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.
s = 6π cm, r = 2 cm, ω = π/4 radian per sec
Verified step by step guidance1
Identify the given variables from the problem: arc length \(s = 6\pi\) cm, radius \(r = 2\) cm, and angular velocity \(\omega = \frac{\pi}{4}\) radians per second.
Recall the formula relating arc length, radius, and angular displacement: \(s = r\theta\). Since \(\theta = \omega t\), substitute to get \(s = r \omega t\).
Rearrange the formula \(s = r \omega t\) to solve for the missing variable \(t\): \(t = \frac{s}{r \omega}\).
Substitute the known values into the equation for \(t\): \(t = \frac{6\pi}{2 \times \frac{\pi}{4}}\).
Simplify the expression step-by-step to find the value of \(t\), which represents the time in seconds.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angular Displacement and Angular Velocity
Angular displacement (θ) measures the angle through which an object rotates, typically in radians. Angular velocity (ω) is the rate of change of angular displacement over time, expressed in radians per second. The relationship ω = θ/t links these quantities, allowing calculation of one if the others are known.
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Introduction to Vectors
Arc Length Formula
The arc length (s) of a circle segment is the distance along the curved path, calculated by s = rθ, where r is the radius and θ is the central angle in radians. This formula connects linear distance traveled along the circumference to angular displacement.
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Combining Angular Velocity and Arc Length
By substituting θ = ωt into s = rθ, the formula becomes s = rωt, linking arc length directly to angular velocity and time. This allows solving for any missing variable (s, r, ω, or t) when the others are known, facilitating problems involving rotational motion.
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4:42Review of Triangles
Related Practice
Textbook Question
Convert each radian measure to degrees. See Examples 2(a) and 2(b). 7π/4
Textbook Question
Find each exact function value. See Example 2.
sec 23π/6
3
views
Textbook Question
The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.
r = 6 cm, ω = π/3 radians per sec, t = 9 sec
Textbook Question
Convert each radian measure to degrees. See Examples 2(a) and 2(b). π/3
3
views
Textbook Question
Find each exact function value. See Example 2.
tan 5π/6