Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). ―315°
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 19
Use the formula ω = θ/t to find the value of the missing variable.
θ = 2π/9 radian , ω = 5π/27 radian per min
Verified step by step guidance1
Identify the given variables and the formula: angular displacement \(\theta = \frac{2\pi}{9}\) radians, angular velocity \(\omega = \frac{5\pi}{27}\) radians per minute, and the formula relating them is \(\omega = \frac{\theta}{t}\), where \(t\) is the time in minutes.
Rearrange the formula to solve for the missing variable \(t\): multiply both sides by \(t\) and then divide both sides by \(\omega\) to isolate \(t\), giving \(t = \frac{\theta}{\omega}\).
Substitute the given values of \(\theta\) and \(\omega\) into the rearranged formula: \(t = \frac{\frac{2\pi}{9}}{\frac{5\pi}{27}}\).
Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \(t = \frac{2\pi}{9} \times \frac{27}{5\pi}\).
Cancel common factors such as \(\pi\) and simplify the numerical fraction to express \(t\) in minutes.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angular Displacement (θ)
Angular displacement represents the angle through which an object rotates, measured in radians. It indicates the change in the angular position of the object and is essential for calculating angular velocity when time is known.
Angular Velocity (ω)
Angular velocity is the rate of change of angular displacement with respect to time, typically expressed in radians per unit time. It quantifies how fast an object rotates and is calculated using the formula ω = θ / t.
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Solving for Time (t) Using ω = θ / t
Given angular displacement and angular velocity, time can be found by rearranging the formula to t = θ / ω. This involves dividing the angular displacement by the angular velocity to determine the duration of rotation.
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