   Chapter 10.3, Problem 51E

Chapter
Section
Textbook Problem

# Show that the curve r = sin θ tan θ (called a cissoid of Diodes) has the line x = 1 as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 0 ≤ x < 1. Use these facts to help sketch the cissoid.

To determine

To find: The polar curve r=sinθtanθ has the line x=1 as vertical asymptote and show all the curve lies in vertical strip 0x<1 .

Explanation

Given:

The polar curve equation r=sinθtanθ

Calculation:

The polar curve is

r=sinθtanθ

The equation for the variable x is given below,

x=rcosθ=sinθtanθcosθ=sin2θ (1)

Take limit as r±

r±sinθtanθ±θ(π2),(π2)+

For θ value (π2) ,

limrx=limθ(π2)sin2θ=1

For θ value (π2)+ ,

limrx=limθ(π2)+sin2θ=1

Hence, limr±x=1 so x=1 is a vertical asymptote

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