Given point Q=(-8√2,8√2) i in rectangular coordinates, what are the corresponding pol
Given point Q=(-8√2,8√2) i in rectangular coordinates, what are the corresponding pol
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°.
Related questions
Question
![### Converting Rectangular Coordinates to Polar Coordinates
**Problem Statement:**
Given point \( Q = (-8\sqrt{2}, 8\sqrt{2}) \) in rectangular coordinates, what are the corresponding polar coordinates?
Rectangular coordinates (also known as Cartesian coordinates) are defined by the pair \((x, y)\), representing points on the x and y axes. Polar coordinates represent points based on their distance \( r \) from the origin and their angle \( \theta \) from the positive x-axis.
**Steps to Convert Rectangular to Polar Coordinates:**
1. **Calculate \( r \) (the radius):**
\[
r = \sqrt{x^2 + y^2}
\]
For \( x = -8\sqrt{2} \) and \( y = 8\sqrt{2} \):
\[
r = \sqrt{(-8\sqrt{2})^2 + (8\sqrt{2})^2} = \sqrt{128 + 128} = \sqrt{256} = 16
\]
2. **Calculate \( \theta \) (the angle):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
For \( x = -8\sqrt{2} \) and \( y = 8\sqrt{2} \):
\[
\frac{y}{x} = \frac{8\sqrt{2}}{-8\sqrt{2}} = -1
\]
The angle whose tangent is \(-1\) is \(-\frac{\pi}{4}\) radians (or \(-45^\circ\)). However, since the point is in the second quadrant (where \( x \) is negative and \( y \) is positive), we adjust the angle by adding \(\pi\):
\[
\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}
\]
Thus, the polar coordinates \( (r, \theta) \) are:
\[
(16, \frac{3\pi}{4})
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcba3479f-edcf-43e4-a58e-fd529b0436e1%2Fa6e41e5d-c915-4059-98bd-92b0a21e2f45%2F9aeih2c_processed.png&w=3840&q=75)
Transcribed Image Text:### Converting Rectangular Coordinates to Polar Coordinates
**Problem Statement:**
Given point \( Q = (-8\sqrt{2}, 8\sqrt{2}) \) in rectangular coordinates, what are the corresponding polar coordinates?
Rectangular coordinates (also known as Cartesian coordinates) are defined by the pair \((x, y)\), representing points on the x and y axes. Polar coordinates represent points based on their distance \( r \) from the origin and their angle \( \theta \) from the positive x-axis.
**Steps to Convert Rectangular to Polar Coordinates:**
1. **Calculate \( r \) (the radius):**
\[
r = \sqrt{x^2 + y^2}
\]
For \( x = -8\sqrt{2} \) and \( y = 8\sqrt{2} \):
\[
r = \sqrt{(-8\sqrt{2})^2 + (8\sqrt{2})^2} = \sqrt{128 + 128} = \sqrt{256} = 16
\]
2. **Calculate \( \theta \) (the angle):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
For \( x = -8\sqrt{2} \) and \( y = 8\sqrt{2} \):
\[
\frac{y}{x} = \frac{8\sqrt{2}}{-8\sqrt{2}} = -1
\]
The angle whose tangent is \(-1\) is \(-\frac{\pi}{4}\) radians (or \(-45^\circ\)). However, since the point is in the second quadrant (where \( x \) is negative and \( y \) is positive), we adjust the angle by adding \(\pi\):
\[
\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}
\]
Thus, the polar coordinates \( (r, \theta) \) are:
\[
(16, \frac{3\pi}{4})
\]
Transcribed Image Text:The image shows a list of four pairs of coordinates written in polar form. Each pair is labeled with a lowercase letter from "a" to "d" on the left side. Here is the detailed transcription of the pairs:
a. \( \left( 8, \frac{3\pi}{4} \right) \)
b. \( \left( 8, \frac{5\pi}{4} \right) \)
c. \( \left( 16, \frac{3\pi}{4} \right) \)
d. \( \left( 16, \frac{5\pi}{4} \right) \)
Explanation:
- Letter 'a' represents the polar coordinate \( \left( 8, \frac{3\pi}{4} \right) \), where the radius is 8 and the angle is \( \frac{3\pi}{4} \) radians.
- Letter 'b' represents the polar coordinate \( \left( 8, \frac{5\pi}{4} \right) \), where the radius is 8 and the angle is \( \frac{5\pi}{4} \) radians.
- Letter 'c' represents the polar coordinate \( \left( 16, \frac{3\pi}{4} \right) \), where the radius is 16 and the angle is \( \frac{3\pi}{4} \) radians.
- Letter 'd' represents the polar coordinate \( \left( 16, \frac{5\pi}{4} \right) \), where the radius is 16 and the angle is \( \frac{5\pi}{4} \) radians.
Polar coordinates are a way of representing points in a plane using the distance from a reference point (called the radius) and an angle from a reference direction (usually the positive x-axis).
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