aflacatn

2021-08-16

Is it true that the equations r=8, ${x}^{2}+{y}^{2}=64$, and $x=8\mathrm{sin}\left(3t\right),y=8\mathrm{cos}\left(3t\right)\left(0\le t\le 2\pi \right)$ all have the same graph.

Aubree Mcintyre

2.Now, we will check that what type of equation is the r=8
This equation is the polar form equation in which we consider $r=\sqrt{{x}^{2}+{y}^{2}}$
So, $r=\sqrt{{x}^{2}+{y}^{2}}=8$
$⇒={x}^{2}+{y}^{2}={8}^{2}=64$
Thus, we can say that after changing polar co-ordinates to Cartesian co-ordinates, equation  r=8  is also the equation of the circle with radius a=8 and center at (0,0).
So, the graphs of the equation
r=8 and ${x}^{2}+{y}^{2}=64$ are same which is given above.
Now, we have the parametric form equations given as:
$x=8\mathrm{sin}\left(3t\right),y=8\mathrm{cos}\left(3t\right),0\le t\le 2\pi$
${x}^{2}=64{\mathrm{sin}}^{2}\left(3t\right)$
and ${y}^{2}=64{\mathrm{cos}}^{2}\left(3t\right)$
$⇒{x}^{2}+{y}^{2}=64{\mathrm{sin}}^{2}\left(3t\right)+64{\mathrm{cos}}^{2}\left(3t\right)$
$=64\left({\mathrm{sin}}^{2}\left(3t\right)+{\mathrm{cos}}^{2}\left(3t\right)\right)=64$
as by trigonometry identity ${\mathrm{sin}}^{2}\left(3t0+{\mathrm{cos}}^{2}\left(3t\right)=1$
thus, we have shown that these parametric equations from circle with radius 8 and center at (0,0).
So, these parametric equations
$x=8\mathrm{sin}\left(3t\right),y=8\mathrm{cos}\left(t\right),0\le t\le 2\pi$ also have the same graph given above.
4. So, the given statement is true as the equation
r=8 ,
${x}^{2}+{y}^{2}=64$ and
$x=8\mathrm{sin}\left(3t\right),y=8\mathrm{cos}\left(t\right),0\le t\le 2\pi$ denote same equation of circle with radius 8 and center at
origin.Thus, all these equations have the same graph given above.