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 given circle is \(2\pi\) square units

Step-by-step solution

Convert the Polar Equation to the Rectangular Coordinate System

The conversion from polar coordinates to rectangular coordinates is done using the relations \(x = r\cos\theta\) and \(y = r\sin\theta\). Substituting \(r = \sin\theta+\cos\theta\) into these relations, we get:\[x=(\sin \theta +\cos \theta)cos\theta = \sin \theta \cos\theta+\cos^2 \theta\]\[y=(\sin \theta +\cos \theta)sin\theta = \sin^2 \theta + \cos \theta\sin\theta \]Squaring and adding these two equations,we get:\[ x^2 + y^2 = (\sin^2 \theta + 2\cos \theta\sin\theta + \cos^2 \theta) = 1 + \sin 2\theta \]

Determine the Radius

Observe that the derived equation corresponds to the square of a typical circle equation in rectangular coordinates \( x^2 + y^2 = r^2 \). From this, we find that the radius is given by \( r = \sqrt{1+\sin 2\theta}\). However, the maximum value of \(\sin 2\theta\) is 1, so the maximum radius is \( r = \sqrt{1+1}= \sqrt{2}\).

Computing the Area

The standard formula for the area of a circle is \(A = \pi r^2\). By substituting \(r = \sqrt{2}\) (the maximum value) into this formula, we get:\[A = \pi (\sqrt{2})^2 = 2\pi\]