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Chapter 4: Problem 9

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). (TABLE CAN NOT COPY)

Short Answer

Expert verified
The radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\) is \(s / r\).

Step by step solution

01

Understand the Definition of a Radian

A radian is defined as the measure of the angle \(\theta\) that, when drawn from the center of a circle, cuts off an arc of length equal to the radius of the circle. In this case, if we let the angular measure of this arc be 1 radian, the length of the arc, \(s\), is equal to the radius, \(r\). In other words, if \(\theta = 1 \text{ radian}\), then \(s = r\).
02

Establish a Proportional Relationship

By the definition of a radian, we know that the number of radians in the central angle is directly proportional to the length of the arc it intercepts. Therefore, if the length of the arc is \(s\) and the measure of the angle in radians is \(\theta\), we can establish the proportion as: \(1 \text{ radian} / r = \theta / s\).
03

Solve for \(\theta\)

From the proportion in Step 2, we can solve for \(\theta\): \(\theta = s / r\). This shows that the radian measure of the central angle of a circle that intercepts an arc of length \(s\) is \(s\) divided by the radius \(r\).

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Most popular questions from this chapter

a. Graph \(y=\tan x\) for \(-\frac{\pi}{2}

If \(f(x)=3 x^{2}-x+5,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) and simplify.

Graph: \(f(x)=\frac{2}{3} x-2.\) (Section 1.4, Example 4).

Explain how to find the radian measure of a central angle.

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to a decimal in degrees. Round your answer to two decimal places. $$30^{015} 10^{\prime \prime}$$
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