Question

Graph each polar equation. Also, identify the type of polar graph.r² = 4 cos 2θ

Accepted Answer

Recognize that the given equation is in polar form: $$r^{2} = 4$$ \cos 2	heta$$. This suggests the graph might be a rose curve or a lemniscate because of the $$\cos 2	heta$$ term and the squared radius. Recall the general form for rose curves: r = a$$ \cos n	heta$$ or r = a$$ \sin n	heta$$. However, since we have r$$^{2}$$ instead of r$, this indicates the graph is a lemniscate, which often has equations like r$$^{2}$$ = a$$^{2}$$ \cos 2	heta or$$ $$r^{2}$$ = $$a^{2}$$ \sin 2	heta$$. To graph the equation, consider values of $$	heta$$ where $$\cos 2	heta$$ is positive or zero, because r$$^{2}$$ must be non-negative. For angles where $$\cos 2	heta$$ is negative, r$$^{2}$$ would be negative, which is not possible for real r$. Calculate r$ for several key angles by substituting values of $$	heta$$ into r$$^{2}$$ = 4 \cos 2	heta$$, then take the positive and negative square roots to find $$r. $$Plot these points in polar coordinates to sketch the graph. Identify the graph type: since the equation matches the form of a lemniscate of Bernoulli (a figure-eight shape), conclude that the graph is a lemniscate centered at the pole with petals aligned along the $$	heta = 0$$ and $$	heta = \pi/2$$ directions.

Graph each polar equation. Also, identify the type of polar graph.
r² = 4 cos 2θ

Risposta video verificata per un problema simile:

Concetti chiave

Polar Coordinates and Equations

Types of Polar Graphs

Trigonometric Identities and Symmetry in Polar Graphs