Chapter 10.4, Problem 101E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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(Î¸) and rotated about the pole through an angle Ï•.

Proof:

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

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

r=g(Î¸)

Consider the diagram given below:

Let (r1,Î¸1) be any point of the curve given as, r=f(Î¸).

Then, (r1,Î¸1+Ï•) is on the curve given as, r=g(Î¸), that is;

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