   Chapter 10.4, Problem 101E

Chapter
Section
Textbook Problem

Rotated Curve Verify that if the curve whose polar equation is r = f ( θ ) is rotated about the pole through an angle ϕ , then an equation for the rotated curve is r = f ( θ − ϕ ) .

To determine

To prove: If the curve whose polar equation is given as, r=f(θ) is rotated about the pole through an angle ϕ, then, the equation for the rotated curve will be r=f(θϕ).

Explanation

Given:

The polar equation is r=f(θ) and rotated about the pole through an angle ϕ.

Proof:

Let the polar equation of the curve be r=f(θ).

Rotate this curve about the pole through the angle ϕ to obtain the following equation of the curve,

r=g(θ)

Consider the diagram given below:

Let (r1,θ1) be any point of the curve given as, r=f(θ).

Then, (r1,θ1+ϕ) is on the curve given as, r=g(θ), that is;

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