Show that if the polar graph of r=f(θ) is rotated counterclockwise around the origin through an

Name: Problem 65, Chapter 10: Calculus Early Transcendentals
Uploaded: 2020-12-11T19:46:04-08:00
Duration: 1 min 44 s
Channel: Carson Merrill
Description: Show that if the polar graph of r=f(θ) is rotated counterclockwise around the origin through an angle α, then r=f(θ-α) is an equation for the rotated curve. [Hint: If (r0, θ0) is any point on the original graph, then (r0, θ0+α) is a point on the rotated graph.]

step 1 So we see what we see with this problem is if we start off with R equals F theta minus, and then

Question

Show that if the polar graph of $r=f(\theta)$ is rotated counterclockwise around the origin through an angle $\alpha$, then $r=f(\theta-\alpha)$ is an equation for the rotated curve. [Hint: If $\left(r_{0}, \theta_{0}\right)$ is any point on the original graph, then $\left(r_{0}, \theta_{0}+\alpha\right)$ is a point on the rotated graph.]

   Show that if the polar graph of $r=f(\theta)$ is rotated counterclockwise around the origin through an angle $\alpha$, then $r=f(\theta-\alpha)$ is an equation for the rotated curve. [Hint: If $\left(r_{0}, \theta_{0}\right)$ is any point on the original graph, then $\left(r_{0}, \theta_{0}+\alpha\right)$ is a point on the rotated graph.]

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 10, Problem 65

Instant Answer

Solved by Expert Carson Merrill

12/11/2020

Step 1

Step 1: Consider a point $(r_0, \theta_0)$ on the original graph $r=f(\theta)$. Show more…

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Show that if the polar graph of $r=f(\theta)$ is rotated counterclockwise around the origin through an angle $\alpha$, then $r=f(\theta-\alpha)$ is an equation for the rotated curve. [Hint: If $\left(r_{0}, \theta_{0}\right)$ is any point on the original graph, then $\left(r_{0}, \theta_{0}+\alpha\right)$ is a point on the rotated graph.]

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Key Concepts

Rotation Transformation in Polar Coordinates

Rotating a graph in polar coordinates involves a transformation of the angle component. Specifically, if a curve described by r = f(?) is rotated counterclockwise by a fixed angle, every point’s angle is increased by that angle, which results in the new equation being r = f(? - ?).

Function Transformation

This concept refers to the process of altering the input or output of a function to achieve a specific transformation, such as a rotation or translation. In the context of polar graphs, substituting ? with (? - ?) in the function f demonstrates how the entire curve is rotated by the angle ?.

Polar Coordinates

This coordinate system represents points in the plane using a distance from a fixed point (the origin) and an angle from a fixed direction. It is commonly used to describe curves that have a naturally circular or angular structure.

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