How can you determine the symmetry of a graph just from the equation r=6sinθ?

Question

I know the symmetry is θ=pi/2 how do I determine this without a calculator just from the equation.

Answer 1

Given: r=6•sinθ

We have to test it in three ways. If the equation is the same as the given after substitution, then the equation is symmetric with respect to the following axis:

(1) Symmetry on polar axis (x-axis): replace θ with -θ

Original: r=6•sinθ

substitution:

r=6•sin(-θ)

r=-6•sin(θ) not the same as the original.

(2) Symmetry on θ=π/2 (y-axis): replace θ with -θ and replace r with -r

substitution:

-r=6•sin(-θ)

-r= -6•sin(θ)

r=6•sin(θ) the same as the original

(3) Symmetry on the pole (r): replace r with -r

Substitution:

-r= 6•sin(θ) not the same as the original

Therefore, it is symmetric with respect to θ=π/2.