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Chapter 4: Problem 48

Most dictionaries define acute angles and obtuse angles in terms of degrees. Restate these definitions in terms of radians.

Short Answer

Expert verified
An acute angle is any angle less than \( \frac{\pi}{2} \) radians. An obtuse angle is any angle between \( \frac{\pi}{2} \) radians and \( \pi \) radians.

Step by step solution

01

Understand the relation between degrees and radians

To convert degrees to radians, we use the following relation: 1 degree = \( \frac{\pi}{180} \) radians.
02

Find the endpoints in radians for acute angles

An acute angle is an angle less than 90 degrees. Using the conversion from Step 1, we can find the value of 90 degrees in radians: 90 degrees = \( 90 \times \frac{\pi}{180} \) = \( \frac{\pi}{2} \) radians. So, an acute angle is an angle less than \( \frac{\pi}{2} \) radians.
03

Find the endpoints in radians for obtuse angles

An obtuse angle is an angle between 90 degrees and 180 degrees. We already found the value of 90 degrees in radians in Step 2, which is \( \frac{\pi}{2} \) radians. Now, we need to find the value of 180 degrees in radians: 180 degrees = \( 180 \times \frac{\pi}{180} \) = \( \pi \) radians. So, an obtuse angle is an angle between \( \frac{\pi}{2} \) radians and \( \pi \) radians. Restating the definitions of acute angles and obtuse angles in terms of radians: An acute angle is any angle less than \( \frac{\pi}{2} \) radians. An obtuse angle is any angle between \( \frac{\pi}{2} \) radians and \( \pi \) radians.

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