Suppose a point has polar coordinates -2, п 37), 2 with the angle measured in radians. Find two additional polar representations of the point. Write each coordinate in simplest form with the angle in [-2π, 2π].
Suppose a point has polar coordinates -2, п 37), 2 with the angle measured in radians. Find two additional polar representations of the point. Write each coordinate in simplest form with the angle in [-2π, 2π].
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Understanding Polar Coordinates and Representations
#### Problem Statement:
Suppose a point has polar coordinates \( \left( -2, \frac{3\pi}{2} \right) \), with the angle measured in radians.
Find two additional polar representations of the point. Write each coordinate in the simplest form with the angle in the interval \([-2\pi, 2\pi]\).
#### Explanation:
Polar coordinates are a pair of values \((r, \theta)\) where:
- \(r\) is the radial distance from the origin.
- \(\theta\) is the angular displacement from the positive x-axis, measured in radians.
Given Coordinates: \( \left( -2, \frac{3\pi}{2} \right) \)
To find additional representations, consider:
1. Adjusting the angle \(\theta\) by adding or subtracting \(2\pi\) to produce equivalent positions on the polar plane.
2. Changing the sign of \(r\) and adjusting the angle \(\theta\) by \(\pi\).
#### Visual Representation:
The image consists of a polar coordinate representation where:
- Two circles represent possible locations on the plane.
- Controls for converting and adjusting these coordinates.
#### Solutions Approach:
1. **First Representation:**
Adjust the angle while maintaining the radius:
- Original: \( \left( -2, \frac{3\pi}{2} \right) \)
- Adding \(2\pi\) to the angle: \(\frac{3\pi}{2} + 2\pi = \frac{7\pi}{2}\)
- Since \(\frac{7\pi}{2}\) exceeds \(2\pi\), convert to an equivalent angle within the given range.
Simplification: \(\frac{7\pi}{2} - 2\pi = \frac{3\pi}{2} + \frac{4\pi}{2}\)
- New Coordinates: \( \left( -2, \frac{3\pi}{2} \right) \)
2. **Second Representation:**
Change the sign of \(r\) and add \(\pi\) to the angle:
- Original: \( \left( -2, \frac{3\pi}{2} \right) \)
- Adjusted: \( \left( 2, \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a948ce4-bfec-4c4b-a693-d4721e6497c9%2F729ee618-9a23-4b4f-8b1e-88c3872857a9%2Fksskqx.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Polar Coordinates and Representations
#### Problem Statement:
Suppose a point has polar coordinates \( \left( -2, \frac{3\pi}{2} \right) \), with the angle measured in radians.
Find two additional polar representations of the point. Write each coordinate in the simplest form with the angle in the interval \([-2\pi, 2\pi]\).
#### Explanation:
Polar coordinates are a pair of values \((r, \theta)\) where:
- \(r\) is the radial distance from the origin.
- \(\theta\) is the angular displacement from the positive x-axis, measured in radians.
Given Coordinates: \( \left( -2, \frac{3\pi}{2} \right) \)
To find additional representations, consider:
1. Adjusting the angle \(\theta\) by adding or subtracting \(2\pi\) to produce equivalent positions on the polar plane.
2. Changing the sign of \(r\) and adjusting the angle \(\theta\) by \(\pi\).
#### Visual Representation:
The image consists of a polar coordinate representation where:
- Two circles represent possible locations on the plane.
- Controls for converting and adjusting these coordinates.
#### Solutions Approach:
1. **First Representation:**
Adjust the angle while maintaining the radius:
- Original: \( \left( -2, \frac{3\pi}{2} \right) \)
- Adding \(2\pi\) to the angle: \(\frac{3\pi}{2} + 2\pi = \frac{7\pi}{2}\)
- Since \(\frac{7\pi}{2}\) exceeds \(2\pi\), convert to an equivalent angle within the given range.
Simplification: \(\frac{7\pi}{2} - 2\pi = \frac{3\pi}{2} + \frac{4\pi}{2}\)
- New Coordinates: \( \left( -2, \frac{3\pi}{2} \right) \)
2. **Second Representation:**
Change the sign of \(r\) and add \(\pi\) to the angle:
- Original: \( \left( -2, \frac{3\pi}{2} \right) \)
- Adjusted: \( \left( 2, \
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