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 $.

Genesis Terrell

Beginner2023-03-27Added 12 answers

Calculating the symmetrical of the graph:

Test for symmetrical :

If $f(r,\theta )=f(-r,-\theta )$, symmetrical to the pole or the origin

If $f(r,\theta )=f(r,-\theta )$, symmetrical to the polar axis or the $x$ axis

If $f(r,\theta )=f(-r,\theta )$, symmetrical to the $y$axis.

Step-1: About the $x$- axis:

$f(r,\theta )\Rightarrow r=4\mathrm{cos\theta}f(r,-\theta )\Rightarrow r=4\mathrm{cos}(-\theta )=4\mathrm{cos}\theta \left[\mathrm{cos}\right(-\theta )=\mathrm{cos\theta}]\therefore f(r,\theta )=f(r,-\theta )$

Now, the graph is symmetrical about the $x$-axis:

Step-2: About origin:

$f(r,\theta )\Rightarrow r=4\mathrm{cos\theta}f(-r,-\theta )\Rightarrow -r=4\mathrm{cos}(-\theta )\Rightarrow r=-4\mathrm{cos}\theta \left[\mathrm{cos}\right(-\theta )=\mathrm{cos\theta}]\therefore f(r,\theta )\ne f(r,-\theta )$

Thus the graph is not symmetrical about origin.

Step-3: About $y$ axis:

$f(r,\theta )\Rightarrow r=4\mathrm{cos\theta}f(-r,\theta )\Rightarrow -r=4\mathrm{cos}\left(\theta \right)\Rightarrow r=-4\mathrm{cos}\theta \therefore f(r,\theta )\ne f(-r,\theta )$

Therefore, the graph is not symmetrical $y$-axis.

Hence, the graph is symmetrical about the $x$-axis only.

Test for symmetrical :

If $f(r,\theta )=f(-r,-\theta )$, symmetrical to the pole or the origin

If $f(r,\theta )=f(r,-\theta )$, symmetrical to the polar axis or the $x$ axis

If $f(r,\theta )=f(-r,\theta )$, symmetrical to the $y$axis.

Step-1: About the $x$- axis:

$f(r,\theta )\Rightarrow r=4\mathrm{cos\theta}f(r,-\theta )\Rightarrow r=4\mathrm{cos}(-\theta )=4\mathrm{cos}\theta \left[\mathrm{cos}\right(-\theta )=\mathrm{cos\theta}]\therefore f(r,\theta )=f(r,-\theta )$

Now, the graph is symmetrical about the $x$-axis:

Step-2: About origin:

$f(r,\theta )\Rightarrow r=4\mathrm{cos\theta}f(-r,-\theta )\Rightarrow -r=4\mathrm{cos}(-\theta )\Rightarrow r=-4\mathrm{cos}\theta \left[\mathrm{cos}\right(-\theta )=\mathrm{cos\theta}]\therefore f(r,\theta )\ne f(r,-\theta )$

Thus the graph is not symmetrical about origin.

Step-3: About $y$ axis:

$f(r,\theta )\Rightarrow r=4\mathrm{cos\theta}f(-r,\theta )\Rightarrow -r=4\mathrm{cos}\left(\theta \right)\Rightarrow r=-4\mathrm{cos}\theta \therefore f(r,\theta )\ne f(-r,\theta )$

Therefore, the graph is not symmetrical $y$-axis.

Hence, the graph is symmetrical about the $x$-axis only.

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