Question

In Exercises $35-50,$ graph each equation. $$r=2+3 \sin \theta$$

   In Exercises $35-50,$ graph each equation.
$$r=2+3 \sin \theta$$

Precalculus

Cynthia Young 2nd Edition

Chapter 7, Problem 50

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Solved by Expert Robert Leedy

06/16/2020

Step 1

The equation $r=a+b\sin\theta$ or $r=a+b\cos\theta$ represents a circle with center at $(a,0)$ and radius $|b|$ in polar coordinates. Here, $a=2$ and $b=3$. Show more…

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In Exercises $35-50,$ graph each equation. $$r=2+3 \sin \theta$$

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Key Concepts

Polar Coordinates

The polar coordinate system represents points on a plane using a distance from a fixed point (the pole) and an angle from a fixed direction. This system is particularly suitable for curves that have circular or rotational symmetry, which makes it useful for representing and graphing curves defined by equations involving trigonometric functions.

Polar Equations

Polar equations express the relationship between the radius r and the angle ?. These equations allow one to generate curves by plugging in different values of ? and calculating the corresponding r, leading to the formation of curves that might be difficult to describe using Cartesian coordinates.

Graphing Techniques in Polar Coordinates

Graphing a polar equation involves evaluating the equation for various angles, determining key points and symmetry, and plotting these on a polar grid to form the complete curve. Recognizing symmetry and typical shapes, such as limacons or cardioids, helps streamline the graphing process.

Limacon

A limacon is a type of polar curve generally given by an equation in the form r = a + b sin? or r = a + b cos?. Depending on the ratio of a to b, limacons can exhibit various shapes such as dimpled forms, convex curves, or curves with an inner loop, making them an important example of polar graphs.

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