Question

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

Short answer

The area of the circle given by the equation \(r=\sin \theta+\cos \theta\) is \(\pi\).

Step-by-step solution

Converting to Rectangular Coordinates

In order to convert from polar to rectangular coordinates, we need to use the relations \(x=r\cos\theta\) and \(y=r\sin\theta\). Substituting \(r=\sin\theta + \cos\theta\) into these, we get \(x=(\sin\theta + \cos\theta)\cos\theta\) and \(y=(\sin\theta + \cos\theta)\sin\theta\)

Identifying the Circle Equation

These can be simplified to \(x=\sin\theta\cos\theta + \cos^2\theta\) and \(y=\sin^2\theta + \sin\theta\cos\theta\). Now find \(x^2 + y^2\): \((\sin\theta\cos\theta + \cos^2\theta)^2 + (\sin^2\theta + \sin\theta\cos\theta)^2\). Simplifying this expression, we obtain the standard form of the equation of a circle \(x^2 + y^2 = 1\), with center at the origin and radius of 1.

Finding the Area of the Circle

The formula for the area of a circle is given by \(\pi r^2\), where \(r\) is the radius. Since the radius is 1, the area of the circle is \(\pi (1^2) = \pi\)