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Chapter 6: Problem 49

Convert the polar equation to rectangular form. $$r=4 \sin \theta$$

Short Answer

Expert verified
The rectangular form of the given polar equation \(r = 4 \sin \theta\) is \(y = 4y\).

Step by step solution

01

Identify the Polar Equation

The given polar equation is \(r = 4 \sin \theta\).
02

Substitute the Polar Variables

Substitute \(y = r \sin \theta\) into the equation.
03

Simplify the Equation

On simplification, it comes out to be \(y = 4r \sin \theta^2\). However, this involves both r and y, which is not suitable for an equation in rectangular form.
04

Substitute r

To eliminate r, use the relation \(r = \sqrt{x^2 + y^2}\) to substitute r in the equation yielding: \(y = 4 \sqrt{x^2 + y^2} * \sin \theta\).
05

Use the Identified Relation

Remember that \( \sin \theta = y/r = y/\sqrt{x^2 + y^2}\) and substitute \(\sin \theta\) in the equation.
06

Simplify the Equation

The equation then simplifies to: \(y = 4y\). Thus, the rectangular form of the given polar equation is \(y = 4y\).

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Most popular questions from this chapter

In Exercises 69 and 70, find the standard form of the equation of the ellipse with the given characteristics. Then sketch the ellipse. Vertices: \((-4,2),(2,2)\); minor axis of length 4

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (1,-\pi / 2) \\ \end{array} $$

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \cos (u-v) $$

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \sin (u-v) $$

True or False? In Exercises 59-61, determine whether the statement is true or false. Justify your answer. The graph of $$ r=\frac{4}{-3-3 \sin \theta} $$ has a horizontal directrix above the pole.
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