Plotting and Converting Polar Coordinates (-6, 3π)

Math Problem Statement

User responses cleared Homework:10.1 HW - Polar Coordinates Question 15, 10.1.37 Part 2 of 4 HW Score: 43.75%, 14 of 32 points Points: 0 of 1

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The polar axis and a ray from the pole that extends horizontally to the left form an angle. The ray is labeled from 0 to 10 in increments of 2. An arc that completes 1 rotation and continues to the ray indicates an angle drawn counterclockwise from the polar axis to the ray. Upper A second ray extending in the opposite direction of the first ray represents negative r dash values and also has tick marks in increments of 2. A point is plotted on the second ray 3 units from the pole. Your answer is correct.C. O

The polar axis and a ray from the pole that extends vertically down form an angle. The ray is labeled from 0 to 10 in increments of 2. An arc that completes 1 rotation and continues to the ray indicates an angle drawn counterclockwise from the polar axis to the ray. Upper A second ray extending in the opposite direction of the first ray represents negative r dash values and also has tick marks in increments of 2. A point is plotted on the second ray 3 units from the pole. Part 2 (a) For the same point on the graph, find new values of left parenthesis r comma theta right parenthesis for which r greater than 0 comma negative 2 pi less than or equals theta less than 0. enter your response here (Type an ordered pair. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression. Simplify your answer.) Choose the graph that correctly plots the point left parenthesis negative 6 comma 3 pi right parenthesis.(a) For the same point on the graph, find new values of left parenthesis r comma theta right parenthesis for which r greater than 0 comma negative 2 pi less than or equals theta less than 0.(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression. Simplify your answer.) Ask my instructor

Solution

Let's break down the problem step by step. You're given the point $(-6, 3\pi)$ in polar coordinates and asked to find different polar coordinate representations for the same point.

1. Plotting the Point $(-6, 3\pi)$

In polar coordinates, the point $(r, \theta)$ means:

$r = -6$ is the distance from the origin (pole). A negative $r$ means the point is in the direction opposite to the angle $\theta$ .
$\theta = 3\pi$ indicates the angle measured counterclockwise from the positive polar axis. Since $3\pi = \pi + 2\pi$ , it is equivalent to rotating to the angle $\pi$ , or 180 degrees (to the left), and then moving in the opposite direction due to the negative $r$ .

Part (a)

You need to find a new polar representation where:

$r > 0$
$-2\pi \leq \theta < 0$

Strategy:

To get $r > 0$ , we change the sign of $r$ and adjust $\theta$ by adding or subtracting $\pi$ .

For the point $(-6, 3\pi)$ , if we want $r > 0$ , we change $r = 6$ . To keep the point the same, we subtract $\pi$ from the angle, which gives us:

$\theta = 3\pi - \pi = 2\pi$

However, this angle $2\pi$ is not in the range $-2\pi \leq \theta < 0$ . To get an equivalent angle in the range, we subtract $2\pi$ from $\theta = 2\pi$ :

$\theta = 2\pi - 2\pi = 0$

So, the coordinates are:

$(6, 0)$

Final Answer for Part (a):

The polar coordinates of the point with $r > 0$ and $-2\pi \leq \theta < 0$ are:

$(6, -\pi)$

This satisfies the required conditions.

Would you like more details on any part of this explanation or have any questions?

Relative Questions:

How do you convert between polar and rectangular coordinates?
What is the significance of a negative $r$ in polar coordinates?
How do you find equivalent angles in polar coordinates?
What are the steps for finding polar coordinates with a positive $r$ ?
How do you handle angles outside the standard range in polar coordinates?

Tip: When working with polar coordinates, always remember to adjust angles carefully when changing the sign of $r$ .

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Math Problem Analysis

Mathematical Concepts

Polar Coordinates
Angle Conversion
Plotting Points in Polar System

Formulas

θ = θ ± 2π
Polar Coordinates Formula (r, θ)

Theorems

Polar Coordinate Theorem
Angle Addition/Subtraction

Suitable Grade Level

Grades 10-12

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