Problem 29 - Essential Calculus

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=\sin \theta $$

Step-by-Step Solution

Answer

The rectangular form of the equation is \(y = r\). The graph of this equation in polar form is a semi-circle in the right half of the coordinate system.

Step 1: Convert the Polar Equation to Rectangular Form

To convert from polar coordinates \((r,\theta)\) to rectangular coordinates \((x,y)\), we need to use the relationships \(r^2 = x^2 + y^2\) and \(y = r \sin \theta\). As the equation is \(r=\sin \theta\), the conversion will give \(r = r \sin \theta\). This simplifies, through division by \(r\) (assuming \(r \neq 0\)), to \(1 = \sin \theta\). Also, by substituting \(y/r\) for \(\sin \theta\), we get \(1 = y/r\) which further gives the rectangular equation as \(y = r\).

Step 2: Sketch the graph

According to the rectangular equation obtained, \(y = r\), the curve is not a usual function in rectangular coordinates because it doesn't represent \(y\) as a function of \(x\). It represents a whole circle centered at the origin. But in the polar form of \(r=\sin\theta\), we observe that for each \(\theta\), the radius \(r\) is non-negative. So, it only represents the right semi-circle from \((0,0)\) to \((0,2)\).

Step 3: Completion and Review

Upon reviewing the process, we see the conversion from polar to rectangular form has been correctly achieved and the graph accurately represents our final equation. This graph shows that the rectangular equation does not fully capture the nature of the polar equation, as it can only represent \(y\) as a function of \(x\) while the polar equation \(r = \sin \theta\) represents a semi-circle.