Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

Step-by-Step Solution

Answer

After following the described steps to convert the polar equation to rectangular form, you would get a resulting equation matching the rectangular coordinate system.

Step 1: Substitute sin(theta) with y/r in the given equation

The sine of the angle \(\theta\) (sin\(\theta\)) in polar coordinates can be written as \(y\) divided by \(r\) (\(y\)/\(r\)) in rectangular coordinates. So, substitute \(y\)/\(r\) for sin\(\theta\) in the equation. The equation becomes \(r = \frac{6}{2 - 3\frac{y}{r}}\).

Step 2: Rewrite equation to eliminate the fraction

Rewrite the equation to remove fractions. Multiplying both sides by \(r\) and \(2 - 3\frac{y}{r}\), the equation becomes \(r^2= 6r(2 - 3\frac{y}{r})\).

Step 3: Replace r^2 with x^2 + y^2

In terms of rectangular coordinates, \(r^2\) equals \(x^2 + y^2\). Replace \(r^2\) in the equation, \(x^2 + y^2 = 6r(2 - 3\frac{y}{r})\).

Step 4: Replace r with sqrt(x^2 + y^2)

In rectangular coordinates, \(r = \sqrt{x^2 + y^2}\). Replace \(r\) in the equation with \(\sqrt{x^2 + y^2}\). The equation becomes \(x^2 + y^2 = 6\sqrt{x^2 + y^2}(2 - \frac{3y}{\sqrt{x^2 + y^2}})\).

Step 5: Solve the equation

Simplify the equation to convert completely to rectangular form. This involves multiplying out the bracket, simplifying fractions where possible and finally, rearranging the equation in terms of y.