### 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:\[(16, \frac{3\pi}{4})\] 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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Answered: Given point Q=(-8√2,8√2) i in…

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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°.

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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:
\[
(16, \frac{3\pi}{4})
\]

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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