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Exercises Trigonometry

Example 6 from Algebra and Trigonometry, 10.4 Polar Coordinates: Graphs

Sketching the Graph of an Inner-Loop Limaçon

Sketch the graph of $r = 2 + 5 cos θ .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Testing for symmetry, we find that the graph of the equation is symmetric about the polar axis. Next, finding the zeros reveals that when $r = 0,$ $θ = 1.98.$ The maximum $| r |$ is found when $\cos θ = 1$ or when $θ = 0.$ Thus, the maximum is found at the point (7, 0).
Even though we have found symmetry, the zero, and the maximum, plotting more points will help to define the shape, and then a pattern will emerge.

See Table 6.

$θ$ $0$ $\frac{π}{6}$ $\frac{π}{3}$ $\frac{π}{2}$ $\frac{2 π}{3}$ $\frac{5 π}{6}$ $π$ $\frac{7 π}{6}$ $\frac{4 π}{3}$ $\frac{3 π}{2}$ $\frac{5 π}{3}$ $\frac{11 π}{6}$ $2 π$

$r$ 7 6.3 4.5 2 −0.5 −2.3 −3 −2.3 −0.5 2 4.5 6.3 7

Table 6

As expected, the values begin to repeat after $θ = π .$ The graph is shown in Figure 12.

Figure 12 Inner-loop limaçon

How did it go?

PreviousSketching the Graph of a One-Loop Limaçon Next Sketching the Graph of a Lemniscate

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.