Is it true that the equations r=8, x^2+y^2=64, and x=8sin(3t), y=8cos(3t) (0<=t<=2pi) all have the same graph.

aflacatn 2021-08-16 Answered
Is it true that the equations r=8, \(\displaystyle{x}^{{2}}+{y}^{{2}}={64}\), and \(\displaystyle{x}={8}{\sin{{\left({3}{t}\right)}}},{y}={8}{\cos{{\left({3}{t}\right)}}}{\left({0}\le{t}\le{2}\pi\right)}\) all have the same graph.

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Expert Answer

Aubree Mcintyre
Answered 2021-08-17 Author has 8283 answers

image
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 \(\displaystyle{r}=\sqrt{{{x}^{{2}}+{y}^{{2}}}}\)
So, \(\displaystyle{r}=\sqrt{{{x}^{{2}}+{y}^{{2}}}}={8}\)
\(\displaystyle\Rightarrow={x}^{{2}}+{y}^{{2}}={8}^{{2}}={64}\)
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 \(\displaystyle{x}^{{2}}+{y}^{{2}}={64}\) are same which is given above.
Now, we have the parametric form equations given as:
\(\displaystyle{x}={8}{\sin{{\left({3}{t}\right)}}},{y}={8}{\cos{{\left({3}{t}\right)}}},{0}\le{t}\le{2}\pi\)
\(\displaystyle{x}^{{2}}={64}{{\sin}^{{2}}{\left({3}{t}\right)}}\)
and \(\displaystyle{y}^{{2}}={64}{{\cos}^{{2}}{\left({3}{t}\right)}}\)
\(\displaystyle\Rightarrow{x}^{{2}}+{y}^{{2}}={64}{{\sin}^{{2}}{\left({3}{t}\right)}}+{64}{{\cos}^{{2}}{\left({3}{t}\right)}}\)
\(\displaystyle={64}{\left({{\sin}^{{2}}{\left({3}{t}\right)}}+{{\cos}^{{2}}{\left({3}{t}\right)}}\right)}={64}\)
as by trigonometry identity \(\displaystyle{{\sin}^{{2}}{\left({3}{t}{0}+{{\cos}^{{2}}{\left({3}{t}\right)}}={1}\right.}}\)
thus, we have shown that these parametric equations from circle with radius 8 and center at (0,0).
So, these parametric equations
\(\displaystyle{x}={8}{\sin{{\left({3}{t}\right)}}},{y}={8}{\cos{{\left({t}\right)}}},{0}\le{t}\le{2}\pi\) also have the same graph given above.
4. So, the given statement is true as the equation 
r=8 ,
\(\displaystyle{x}^{{2}}+{y}^{{2}}={64}\) and
\(\displaystyle{x}={8}{\sin{{\left({3}{t}\right)}}},{y}={8}{\cos{{\left({t}\right)}}},{0}\le{t}\le{2}\pi\) denote same equation of circle with radius 8 and center at
origin.Thus, all these equations have the same graph given above.
Answer: it`s TRUE

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