Question

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 5 cos 3θ

Answer 1

Well, if you have symmetry about the x-axis, then replacing θ with -θ will give you the same function. The cosine and secant functions are both even functions, meaning that cos-θ = cosθ and sec-θ = secθ. So in this case it is symmetric about the x-axis. Just to check the other two and make sure we know how to test, we can do that. y-axis symmetry is when (r,θ) = (-r, -θ). So we have -r = 5cos-3θ. This does not equal our original problem, so we do not have y-axis symmetry. For origin symmetry, we want to see if r = -r. So we have -r = 5cos3θ, we do not come up with our original. So there is only c-axis symmetry.

Answer 2

*x-axis

Answer 3

let's test it for x-axis symmetry, as @Psymon pointed out, we'll replace the angle with a negative value and see if we end up with the same original function

\(\bf r = 5 cos(3-\theta) \implies r = 5 cos(-3\theta)\\
\text{we know that } cos(-a) = cos(a)\\
r = 5 cos(-3\theta) \implies r = 5 cos(3-\theta)\)

so it's symmetric about the x-axis

Answer 4

I still am inefficient with actually typing up problems using the draw feature, so that definitely makes it look better : )

Answer 5

Thank you. You two really helped me out

Answer 6

@Psymon there are several good LaTex editors, heck is what I use, I don't use the buttons provided here hehehe

Answer 7

\(\bf r = 5 cos(3-\theta) \implies r = 5 cos(-3\theta)\\
\text{we know that } cos(-a) = cos(a)\\
r = 5 cos(-3\theta) \implies r = 5 cos(3\theta)\) I meant

Answer 8

had a typo, but anyhow

Answer 9

so now let's test about the y-axis
again as suggested by @Psymon , we'll replace the "r" and angle with negative values both
\(\bf -r = 5 cos(3(-\theta)) \implies -r = 5 cos(-3\theta)\\
\text{we know that } cos(-a) = cos(a)\\
-r = 5 cos(-3\theta) \implies -r = 5 cos(3\theta)\\
r = -5 cos(3\theta)\)

so the resultant equation doesn't RESEMBLE the original one, thus there's no symmetry about the y-axis

Answer 10

now to test for the origin, just replace "r" for negative or "-r"

Answer 11

see if you get the the same original after factoring and simplifying, if not, then no dice, then again if you read Psymon's line above, he already did :)