Assume the following angle is in standard position. Find a positive and negative angle whose absolute value is less than 2 and is coterminal with the given angle: 8л
Assume the following angle is in standard position. Find a positive and negative angle whose absolute value is less than 2 and is coterminal with the given angle: 8л
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter1: The Six Trigonometric Functions
Section1.2: The Rectangular Coordinate System
Problem 90PS: Draw 60 in standard position. Then find b if the point (2, b) is on the terminal side of 60.
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Question
![**Problem Statement:**
Assume the following angle is in standard position. Find a positive and negative angle whose absolute value is less than \(2\pi\) and is coterminal with the given angle: \(\frac{-8\pi}{3}\).
**Instructions for Students:**
- Use the paperclip button below to attach files.
- *Student can enter max 2000 characters*
**Answer Box:**
This setup provides space where students can write or paste their answers. The problem involves finding angles coterminal with \(\frac{-8\pi}{3}\).
**Detailed Explanation of Concepts:**
When given an angle like \(\frac{-8\pi}{3}\), coterminal angles are those that share the same initial and terminal sides. To find coterminal angles, we add or subtract multiples of \(2\pi\):
1. First, convert \(\frac{-8\pi}{3}\) to a more familiar range by adding \(2\pi\). Since one full rotation is \(2\pi\):
\[
\frac{-8\pi}{3} + 2\pi = \frac{-8\pi}{3} + \frac{6\pi}{3} = \frac{-2\pi}{3}
\]
2. Check the range of the result:
\[
\frac{-2\pi}{3}
\]
This angle is already in the range of \(-2\pi\) to \(2\pi\). Therefore, \(\frac{-2\pi}{3}\) is coterminal with \(\frac{-8\pi}{3}\).
3. For a positive coterminal angle, continue adding \(2\pi\) until the result is positive:
\[
\frac{-2\pi}{3} + 2\pi = \frac{-2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3}
\]
So, the angles \(\frac{-2\pi}{3}\) and \(\frac{4\pi}{3}\) are coterminal with \(\frac{-8\pi}{3}\), with \(\frac{-2\pi}{3}\) being the negative angle, and \(\frac{4\pi}{3}\) being the positive angle within the specified range](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9fc99ab-b3a9-4577-a5d7-bccfea12d417%2F81d6335d-8d9a-4414-a253-c1d6f2fec313%2Fa4apqal_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Assume the following angle is in standard position. Find a positive and negative angle whose absolute value is less than \(2\pi\) and is coterminal with the given angle: \(\frac{-8\pi}{3}\).
**Instructions for Students:**
- Use the paperclip button below to attach files.
- *Student can enter max 2000 characters*
**Answer Box:**
This setup provides space where students can write or paste their answers. The problem involves finding angles coterminal with \(\frac{-8\pi}{3}\).
**Detailed Explanation of Concepts:**
When given an angle like \(\frac{-8\pi}{3}\), coterminal angles are those that share the same initial and terminal sides. To find coterminal angles, we add or subtract multiples of \(2\pi\):
1. First, convert \(\frac{-8\pi}{3}\) to a more familiar range by adding \(2\pi\). Since one full rotation is \(2\pi\):
\[
\frac{-8\pi}{3} + 2\pi = \frac{-8\pi}{3} + \frac{6\pi}{3} = \frac{-2\pi}{3}
\]
2. Check the range of the result:
\[
\frac{-2\pi}{3}
\]
This angle is already in the range of \(-2\pi\) to \(2\pi\). Therefore, \(\frac{-2\pi}{3}\) is coterminal with \(\frac{-8\pi}{3}\).
3. For a positive coterminal angle, continue adding \(2\pi\) until the result is positive:
\[
\frac{-2\pi}{3} + 2\pi = \frac{-2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3}
\]
So, the angles \(\frac{-2\pi}{3}\) and \(\frac{4\pi}{3}\) are coterminal with \(\frac{-8\pi}{3}\), with \(\frac{-2\pi}{3}\) being the negative angle, and \(\frac{4\pi}{3}\) being the positive angle within the specified range
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