Question

Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously

Short answer

Finding intersections for polar graphs might need further analysis beyond simultaneous equations because an individual point can have different representations in the polar coordinate system. It requires verification for both angles and radius for each solution to ensure that they adhere to both polar equations.

Step-by-step solution

Understanding Polar Coordinates

Instead of the cartesian coordinate system which uses x and y coordinates system to plot any point, a polar coordinate system plots a point using radius (or distance from origin) and angle. A location can be represented by multiple representations, e.g., the point at (1, \( \pi \)) can also be represented as (-1, 0).

Points of intersection on Cartesian planes versus Polar planes

In cartesian planes, points of intersection are usually found by solving two equations together. In polar coordinates, the same process can produce variations due to multiple representations. For instance, changing \(\theta\) by \(2\pi\) gives a different representation.

Explaining the need for additional analysis for Polar planes

The multiple representations, one will require further analysis beyond simply solving two equations simultaneously to find the points of intersection on polar graphs. It demands verification of each solution’s radius and angle to guarantee it coincides with both equations' polar curves.

Illustration with an example

Let's have two polar equations r = cos(\(\theta\)) and r = sin(\(\theta\)). The simultaneous equations will give us \(\theta\) = \(\frac{\pi}{4}\), \(\frac{5\pi}{4}\) for r = \(\frac{\sqrt{2}}{2}\). By plugging in the values of \(\theta\) in the first equation and with analysis, we see that \(\theta\) = \(\frac{\pi}{4}\) satisfies both equations, but \(\theta\) = \(\frac{5\pi}{4}\) doesn't, and that's why extra calculation/verification is needed after finding points from simultaneous equations.