Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problem 5

Chapter 4, Problem 5

CONCEPT PREVIEW Find the measure of each central angle (in radians).

Verified step by step guidance

Recall that a central angle in a circle is the angle formed at the center of the circle by two radii.

Understand that the measure of a central angle in radians is related to the arc length it subtends on the circle by the formula: \(\theta = \frac{s}{r}\), where \(\theta\) is the central angle in radians, \(s\) is the arc length, and \(r\) is the radius of the circle.

Identify the given values in the problem: the arc length \(s\) and the radius \(r\) of the circle.

Substitute the known values of \(s\) and \(r\) into the formula \(\theta = \frac{s}{r}\) to express the central angle in radians.

Simplify the fraction to find the measure of the central angle in radians.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Central Angle

A central angle is an angle whose vertex is at the center of a circle and whose sides intersect the circle, forming an arc. The measure of a central angle corresponds directly to the length of the arc it intercepts, making it fundamental in relating angles to arc lengths.

Radian Measure

Radians measure angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. This unit connects linear and angular measurements, with 2π radians equal to 360 degrees, simplifying calculations involving circles.

Arc Length and Angle Relationship

The arc length (s) of a circle is related to the radius (r) and central angle (θ in radians) by the formula s = rθ. Understanding this relationship allows one to find the central angle if the arc length and radius are known, which is essential for solving problems involving central angles.

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Related Practice

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CONCEPT PREVIEW Find the radius of each circle.

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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.

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Convert each degree measure to radians. Leave answers as multiples of π.

175°

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A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.

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CONCEPT PREVIEW Find the measure of each central angle (in radians).

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Textbook Question

Graph each function over a one-period interval.

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