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 6

Chapter 4, Problem 6

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

Verified step by step guidance

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

Recall that the total measure of all central angles around a point (the center of the circle) is \(2\pi\) radians.

If the problem involves dividing the circle into equal parts, determine how many parts the circle is divided into.

Use the formula for each central angle when the circle is divided into \(n\) equal parts: \(\text{Central angle} = \frac{2\pi}{n}\) radians.

Substitute the given number of parts into the formula to express the measure of each 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 are a unit of angular measure based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. This unit simplifies calculations involving circles and is essential for expressing central angles in radians.

Arc Length and Angle Relationship

The measure of a central angle in radians is equal to the length of the intercepted arc divided by the radius of the circle (θ = s/r). Understanding this relationship allows for converting between arc length and angle measure, which is key to solving problems involving central angles.

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Related Practice

Textbook Question

CONCEPT PREVIEW Find the area of each sector.

Textbook Question

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

Textbook Question

CONCEPT PREVIEW Find the radius of each circle.

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

175°

Textbook Question

Graph each function over a one-period interval.

y = - (1/2) csc (x + π/2)

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