Consider the lemniscate given by

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

r^2 = 4 \cos 2\theta

. Throughout this problem restrict attention to the first quadrant (

0<\theta<\tfrac{\pi}{2}

), where the curve is traced exactly once. (a) Express

r

explicitly as a function of

\theta

on the stated domain. (b) Determine the value of

\theta

in the first quadrant at which the point on the lemniscate is farthest from the pole. Justify that this value indeed yields the maximum distance. (c) Give the exact polar coordinate of this farthest point and rewrite it in Cartesian form

(x,y)

. (d) Explain geometrically why no greater value of

r

is possible for any point of the lemniscate outside the first quadrant.

Related theory

Polar Coordinates
Trigonometric Functions
Polar Graphs

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