addilee
contestada

HELP!!! Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 4 cos 3θ

I don't think it's symmetric?

Respuesta :

testing for y-symmetry let's make r = 5 cos 3θ and set θ=π−θ sor=5cos3θθ=(π−θ)r=5cos(3(π−θ))⟹r=5cos(3π−3θ)

Answer:

                The graph is symmetric about the x-axis.

Step-by-step explanation:

  • The graph is symmetric about the polar axis(x-axis) if we replace (r,θ) with (r,-θ) then it is equivalent to the original equation.
  • The graph is symmetric about the y-axis if we replace (r,θ) with (-r,-θ) then it is equivalent to the original equation.
  • The graph is symmetric about the pole(origin) if we replace (r,θ) with (-r,θ) then it is equivalent to the original equation.

Here we have equation as:

           [tex]r=4\cos 3\theta[/tex]

  • when we replace (r,θ) with (r,-θ) we have:

[tex]r=4\cos 3(-\theta)\\\\i.e.\\\\r=4\cos (-3\theta)\\\\i.e.\\\\r=4\cos 3\theta[/tex]

(  Since, we know that:

[tex]\cos (-\tehta)=\cos \theta[/tex]  )

Hence, the graph is symmetric about the x-axis.

  • when we replace  (r,θ) with (-r,-θ)

[tex]-r=4\cos 3(-\theta)\\\\i.e.\\\\-r=4\cos (-3\theta)\\\\i.e.\\\\-r=4\cos 3\theta\\\\i.e.\\\\r=-4\cos 3\theta[/tex]

Hence, we observe that on replacing the function the two graphs are not equivalent.

Hence, the graph is not symmetric about the y-axis.

  • when we replace (r,θ) with (-r,θ) we have:

[tex]-r=4\cos 3\theta\\\\i.e.\\\\r=-4\cos 3\theta[/tex]

Hence, we observe that on replacing the function the two graphs are not equivalent.

Hence, the graph is not symmetric about the pole(origin)

Otras preguntas