Question

Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points. $$r=3 \sin \theta \text { and } r=3 \cos \theta$$

Short answer

Answer: The intersection points of the curves given by $$r=3 \sin \theta$$ and $$r=3 \cos \theta$$ are $$\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$$ and $$\left(-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2}\right)$$.

Step-by-step solution

Set the equations equal to each other

First, in order to find where the two curves intersect, we must set the two equations equal to each other: $$3 \sin \theta = 3 \cos \theta$$.

Isolate the trigonometric functions

To continue solving, we want to isolate the trigonometric functions on one side of the equation. So, we can simply divide both sides by 3 and get the equation: $$\sin \theta = \cos \theta$$.

Use the identity for sine and cosine

We know that the sine and cosine functions are related by the identity $$\sin \left(\frac{\pi}{2} - \theta\right) = \cos \theta$$. So, we can rewrite our equation as $$\sin \theta = \sin \left(\frac{\pi}{2} - \theta\right)$$.

Find the angles where the functions are equal

By the identity, when $$\theta = \frac{\pi}{4}$$, the sine function will be equal to the cosine function. Using the identity $$\sin \theta = \sin \left(\frac{\pi}{2} - \theta\right)$$, we identify $$\theta = \frac{\pi}{4}$$ as one of the angles for which the two polar curves intersect.

Graph the curves

To find the remaining intersection points, we must graph the curves in a polar coordinate system. We know that $$\theta = \frac{\pi}{4}$$ is one of the intersecting angles, and after graphing the two curves, we can also see that they intersect at $$\theta = -\frac{3\pi}{4}$$. So, our intersection points are: $$\left(3\sin(\frac{\pi}{4}), \frac{\pi}{4}\right)$$ and $$\left(3\sin(-\frac{3\pi}{4}), -\frac{3\pi}{4}\right)$$.

Convert polar coordinates to rectangular coordinates

To fully understand the intersection points, we will convert the polar coordinates to rectangular coordinates. $$\left(3\sin(\frac{\pi}{4}), \frac{\pi}{4}\right) = (\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2})$$ and $$\left(3\sin(-\frac{3\pi}{4}), -\frac{3\pi}{4}\right) = (-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2})$$. The intersection points of the curves given by $$r=3 \sin \theta$$ and $$r=3 \cos \theta$$ are $$\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$$ and $$\left(-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2}\right)$$.