Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r = 4\sin(\theta)\) is \(yx^2 = 4x^2 - 4(x^2 + y^2)\), after substituting and simplifying.

Step 1: Review the conversion relation between coordinate systems

The connection between polar and rectangular coordinates is given by \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).

Step 2: Substitute the given polar equation into rectangular form

The given polar equation is \(r = 4\sin(\theta)\). We can substitute \(r\) into the expression for \(y\) as follows: \(y = r\sin(\theta) = 4\sin^2(\theta)\).

Step 3: Reduce the equation

To obtain the most common representation of this equation, use the Pythagorean identity \(1-\cos^2(\theta)=\sin^2(\theta)\). This will give us: \(y = 4(1-\cos^2(\theta))\). Next, substitute \(x = r\cos(\theta)\) into our equation to replace \(\cos(\theta)\). Solving for r, we find that \(r = \frac{x}{\cos(\theta)}\). Substituting this gives: \(y = 4(1-(r/x)^2)\). This needs to be further reduced in order to give x and y in terms of r.

Step 4: Solve the final equation in terms of x and y

To get y in terms of x, multiply both sides by x^2: \(yx^2 = 4x^2 - 4r^2\). Our equation: \(r^2 = x^2 + y^2\), will replace \(r^2\), so finally we get \(yx^2 = 4x^2 - 4(x^2 + y^2)\).