Question

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

Short answer

Finding points of intersection of polar graphs requires more than solving two equations simultaneously due to the presence of multiple representations of the same point.

Step-by-step solution

Understanding Polar Coordinates and Their Complexity

In polar coordinates, a point is represented by two parameters - radius (\( r \)) and angle (\( \theta \)). It's important to understand that, unlike Cartesian coordinates, a single point in polar coordinates is often represented by multiple pairs of (\( r, \theta \)).

Intersecting Points in Polar Coordinates

When two polar graphs intersect, we initially set the two equations equal to each other to find possible intersections. This gives us the initial set of candidates for points of intersection. However, the complexity of multi-representation in polar form means that these initially spotted intersection points may not be the only ones.

Further Investigation of Potential Intersection Points

We perform further analysis by checking all possible representations of these points as polar coordinates. This usually requires testing different values of \( \theta \), as each value of \( \theta \) typically corresponds to more than one value of \( r \). This process verifies if a point is actually a point of intersection or not.

Example

For instance, consider the polar graphs \( r = \cos(\theta) \) and \( r = \sin(2\theta) \). Upon solving these equations together, we get \( \cos(\theta) = \sin(2\theta) \) and can immediately spot \(\theta = 0\) as one of the solutions. This nevertheless isn't the only solution, as \( \theta = \frac{\pi}{4}, \frac{5\pi}{4} \) are also solutions. This shows the need to conduct further analysis beyond solving two equations simultaneously.