Question

How do you test for symmetry for $r=1-2\sin \left( \theta \right)$ ?

Answer 1

In this question we have been asked to test the given expression $r=1-2\sin \theta$ for symmetry. We know that $\sin \theta$ is symmetric about the y-axis. That is $\sin \left( \pi -\theta \right)=\sin \theta$ . The definition of symmetry says that “if $\left( x,y \right)$ is a point on the curve then $\left( p,q \right)$ another point that is equidistant (mirror image) with respect to y-axis (or x-axis) should also lie on the curve.

Complete step by step solution:
Now considering the question we have been asked to test the given expression $r=1-2\sin \theta$ for symmetry.
From the basic concepts of trigonometry we know that $\sin \theta$ is symmetric about the y-axis. That is $\sin \left( \pi -\theta \right)=\sin \theta$ .
From the basic concepts we know that the definition of symmetry says that “if $\left( x,y \right)$ is a point on the curve then $\left( p,q \right)$ another point that is equidistant (mirror image) with respect to y-axis (or x-axis) should also lie on the curve.
Now we can say that for a point $\left( r,\theta \right)$ the equidistant point will be $\left( r,\pi -\theta \right)$ with respect to y-axis. If we verify this point by substituting it in the given expression we will have $\begin{aligned} & \Rightarrow r=1-2\sin \left( \pi -\theta \right) \\\ & \Rightarrow r=1-2\sin \theta \\\ \end{aligned}$ .
Hence we can conclude that the given expression $r=1-2\sin \theta$ is symmetric about the y-axis.
The graph of this curve is shown below:

Note: While answering questions of this type we should be sure with our concepts that we are going to apply. Similarly we also know that $\cos \theta$ is symmetric about the x-axis that is $\cos \left( -\theta \right)=\cos \theta$. Someone can confuse between these two and consider that the sine function is symmetric about the x-axis and end up having a wrong conclusion so we should be careful.