Q. 9 - Calculus

Question

Find all values of $\alpha$ such that the graphs of $\theta =\alpha$ and $\theta =-\alpha$ are the same in a polar coordinate system.

Step-by-Step Solution

Answer

The equations $\theta =\alpha$ and $\theta =-\alpha$ are identical in polar coordinate system. 

Therefore, the answer is $\theta =\alpha \frac{\pi }{2},\alpha$ is any integer

Step 1: Given information

The equations $\theta =\alpha$ and $\theta =-\alpha .$

Step 2: simplification

Consider the equations $\theta =\alpha$ and $\theta =-\alpha .$

The objective is to show that $\theta =\alpha$ and $\theta =-\alpha$ are same in polar coordinate system.

For any real number $\alpha$, the polar equation $\theta =\alpha$ describes the set of points with polar angle $\alpha$.

The angle is positive in counterclockwise direction and negative in the clockwise direction.

When $\theta =\alpha$ the value of $r$ can be either positive or negative.

Thus, the graph for $\theta =\alpha$ is line through the pole.

When $\theta =-\alpha$ the value of $r$ can be either positive or negative. and $\theta =-\alpha$ lies on the same line that of $\theta =\alpha$ which passes through the pole.

Example:

Consider $\theta =\alpha \frac{\pi }{2}$.where $\alpha$ is any integer.

The graphical representation of $\theta =\alpha$ and $\theta =-\alpha$.

That means the equations $\theta =\alpha$ and $\theta =-\alpha$ are identical in polar coordinate system.

Therefore, the answer is $\theta =\alpha \frac{\pi }{2},\alpha$ is any integer