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 4

Chapter 4, Problem 4

CONCEPT PREVIEW Find the radius of each circle.

Verified step by step guidance

Identify the given information about the circle, such as the coordinates of the center and a point on the circle, or the length of a chord, diameter, or any other relevant measurement.

Recall the formula for the radius of a circle. If you know the center \((h, k)\) and a point on the circle \((x, y)\), use the distance formula to find the radius: \[r = \sqrt{(x - h)^2 + (y - k)^2}\]

If the problem provides the diameter \(d\), remember that the radius is half the diameter: \[r = \frac{d}{2}\]

If the problem involves trigonometric relationships (such as angles subtended by chords or arcs), use the appropriate trigonometric formulas to express the radius in terms of the given quantities.

Substitute the known values into the chosen formula and simplify the expression to find the radius. Avoid calculating the final numeric value unless required.

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Key Concepts

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

Definition of a Circle and Radius

A circle is a set of points equidistant from a fixed center point. The radius is the distance from the center to any point on the circle, and it is constant for a given circle. Understanding this helps identify what is being asked when finding the radius.

Using the Distance Formula

If the coordinates of the center and a point on the circle are given, the radius can be found using the distance formula: \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This formula calculates the length between two points in the plane.

Equation of a Circle

The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius. Recognizing or manipulating this equation allows extraction of the radius by comparing terms.

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Related Practice

Textbook Question

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

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

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.

Textbook Question

Find the linear speed v for each of the following.

a point on the equator moving due to Earth's rotation, if the radius is 3960 mi

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Textbook Question

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.

y = - (1/2) csc (x + π/2)

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