Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The rectangular form of the given polar equation is \(y = 2x - 3\)

Step 1: Write down the relations between Polar and Rectangular coordinates

The following relationships exist between polar and rectangular coordinates: \(x = r\cos \theta\) and \(y = r\sin \theta\). These will be used to convert from polar form to rectangular form.

Step 2: Substitute the value of r into the polar to rectangular relations

Use the provided polar equation and substitute \(r=\frac{6}{2\cos \theta-3\sin \theta}\) into both relations from step 1, as follows : \(x = \frac{6\cos \theta}{2\cos \theta-3\sin \theta}\) and \(y = \frac{6\sin \theta}{2\cos \theta-3\sin \theta}\). After simplification you will have: \(x = \frac{6\cos \theta}{2\cos \theta-3\sin \theta}\) and \(y = 2x-3\).

Step 3: Solve for y in the rectangular form

Equating the two relations allows us to solve for y. On equating, we have \(y = 2x-3\).\This is the required rectangular form.