Sure, let's convert the given polar equation to its rectangular form step-by-step:
Given polar equation:
[tex]\[ r - 7 \sin \theta = 0 \][/tex]
To convert this to rectangular form, we need to use the relationships between polar and rectangular coordinates:
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
[tex]\[ \sin \theta = \frac{y}{r} \][/tex]
First, substitute these expressions into the given equation:
[tex]\[ \sqrt{x^2 + y^2} - 7 \left(\frac{y}{\sqrt{x^2 + y^2}}\right) = 0 \][/tex]
Multiply every term by [tex]\(\sqrt{x^2 + y^2}\)[/tex] to clear the fraction:
[tex]\[ (\sqrt{x^2 + y^2})^2 - 7y = 0 \][/tex]
Simplify [tex]\((\sqrt{x^2 + y^2})^2\)[/tex] to [tex]\(x^2 + y^2\)[/tex]:
[tex]\[ x^2 + y^2 - 7y = 0 \][/tex]
Thus, the rectangular form of the given polar equation in general form is:
[tex]\[ x^2 + y^2 - 7y = 0 \][/tex]
So, we have:
[tex]\[
x^2 + y^2 - 7y = 0
\][/tex]
