To convert the given polar equation to its rectangular form, let’s follow a step-by-step approach:
1. Given Polar Equation:
[tex]\[
-15 \csc \theta = 5r
\][/tex]
2. Identify Relationships in Polar Coordinates:
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Therefore, the equation can be rewritten using [tex]\(\csc \theta\)[/tex]:
[tex]\[
-15 \left( \frac{1}{\sin \theta} \right) = 5r
\][/tex]
3. Simplify the Equation:
Simplify the equation by converting [tex]\(\csc \theta\)[/tex]:
[tex]\[
-15 \frac{1}{\sin \theta} = 5r
\][/tex]
Multiply both sides by [tex]\(\sin \theta\)[/tex] to clear the fraction:
[tex]\[
-15 = 5r \sin \theta
\][/tex]
Divide both sides by 5:
[tex]\[
-3 = r \sin \theta
\][/tex]
4. Convert to Rectangular Coordinates:
Recall that in polar coordinates:
[tex]\[
r \sin \theta = y
\][/tex]
Substitute [tex]\(y\)[/tex] in place of [tex]\(r \sin \theta\)[/tex]:
[tex]\[
-3 = y
\][/tex]
5. Rectangular Form of the Equation:
The rectangular form of the equation is:
[tex]\[
y = -3
\][/tex]
6. Express in General Form:
Moving all terms to one side to express it in general form:
[tex]\[
y + 3 = 0
\][/tex]
Therefore, the rectangular form of the given polar equation in general form is:
[tex]\[
y + 3 = 0
\][/tex]
