Ressuli422p

2023-03-26

Determine if the graph is symmetric about the $x$-axis, the $y$-axis, or the origin.$r=4\mathrm{cos}3\theta$.

### Answer & Explanation

Genesis Terrell

Calculating the symmetrical of the graph:
Test for symmetrical :
If $f\left(r,\theta \right)=f\left(-r,-\theta \right)$, symmetrical to the pole or the origin
If $f\left(r,\theta \right)=f\left(r,-\theta \right)$, symmetrical to the polar axis or the $x$ axis
If $f\left(r,\theta \right)=f\left(-r,\theta \right)$, symmetrical to the $y$axis.
Step-1: About the $x$- axis:
$f\left(r,\theta \right)⇒r=4\mathrm{cos\theta }f\left(r,-\theta \right)⇒r=4\mathrm{cos}\left(-\theta \right)=4\mathrm{cos}\theta \left[\mathrm{cos}\left(-\theta \right)=\mathrm{cos\theta }\right]\therefore f\left(r,\theta \right)=f\left(r,-\theta \right)$
Now, the graph is symmetrical about the $x$-axis:
$f\left(r,\theta \right)⇒r=4\mathrm{cos\theta }f\left(-r,-\theta \right)⇒-r=4\mathrm{cos}\left(-\theta \right)⇒r=-4\mathrm{cos}\theta \left[\mathrm{cos}\left(-\theta \right)=\mathrm{cos\theta }\right]\therefore f\left(r,\theta \right)\ne f\left(r,-\theta \right)$
Thus the graph is not symmetrical about origin.
Step-3: About $y$ axis:
$f\left(r,\theta \right)⇒r=4\mathrm{cos\theta }f\left(-r,\theta \right)⇒-r=4\mathrm{cos}\left(\theta \right)⇒r=-4\mathrm{cos}\theta \therefore f\left(r,\theta \right)\ne f\left(-r,\theta \right)$
Therefore, the graph is not symmetrical $y$-axis.
Hence, the graph is symmetrical about the $x$-axis only.

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