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Chapter 6: Problem 59

Convert the polar equation to rectangular form. $$r=2 \sin 3 \theta$$

Short Answer

Expert verified
The rectangular form of the given polar equation \(r = 2 \sin 3 \theta\) is \(3y - 4(y^3/(\sqrt{x^2 + y^2})^2) - \sqrt{x^2 + y^2} = 0\).

Step by step solution

01

Express r using rectangular coordinates

Start by expressing \(r\) in terms of \(x\) and \(y\) using the relationship \(r^2 = x^2 + y^2\). This gives \(r = \sqrt{x^2 + y^2}\).
02

Express sin 3theta using double-angle formulas

Express \(\sin 3 \theta\) using the angle double formulas. This gives \(\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta\). To write \(\sin \theta\), use the relationship \(\sin \theta = y / r\). So, \(3y/r - 4(y/r)^3 = \sqrt{x^2 + y^2}\).
03

Convert the equation to rectangular form

Now, replace \(r\) on the right side of the equation with \(\sqrt{x^2 + y^2}\) from step 1. This gives you the rectangular form of the equation: \(3y - 4(y^3/(\sqrt{x^2 + y^2})^2) - \sqrt{x^2 + y^2} = 0\). You might simplify further as needed.

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Most popular questions from this chapter

\(r=3 \cos 2 \theta\)

The equation $$ r=\frac{e p}{1 \pm e \sin \theta} $$ is the equation of an ellipse with \(e<1\). What happens to the lengths of both the major axis and the minor axis when the value of \(e\) remains fixed and the value of \(p\) changes? Use an example to explain your reasoning.

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Ellipse } & e=\frac{3}{4} & y=-3 \\ \end{array} $$

\(r=5+4 \cos \theta\)

\(r=2 \cos (3 \theta-2)\)
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