Answer:
Please see attached image for the graph
Step-by-step explanation:
Give different values of "t" in radians, selecting those that can facilitate the calculation of sine and cosine. The values need to be in the interval from zero to [tex]3\,\pi[/tex].
Notice that for t = 0 we get :
[tex]x=3\,cos(0)-0+5=3+5=8\\y=sin(0)=0[/tex]
This is our initial point (8,0)
We can evaluate for [tex]t=\frac{\pi}{2}[/tex] which also gives an easy to calculate value:
[tex]x=3\,cos (\frac{\pi}{2} )-\frac{\pi}{2} +5=3*0-\frac{\pi}{2} +5=5-\frac{\pi}{2} \\y=sin(\frac{\pi}{2} )=1[/tex]
This point is the maximum reached by the curve on the first quadrant.
Another easy point is for [tex]t=\pi[/tex], which gives:
[tex]x=3\,cos(\pi)-\pi+5=-3\,-\pi +5=2-\pi\\y=sin(\pi)=0[/tex]
This is the crossing of the x-axis that we see to the left of the origin of coordinates.
The crossing of the x axis that appears then on the right of the origin of coordinates is when we use [tex]t=2\,\pi[/tex] (another simple to calculate point)
The minimum value reached by the graph (on the fourth quadrant) is obtained when we use [tex]t=\frac{3\,\pi}{2}[/tex]
The maximum observed for the graph on the second quadrant is obtained when we use [tex]t=\frac{5\,\pi}{2}[/tex].
And finally, the last point we obtained is when [tex]t=3\, \pi[/tex] which is the point where the graph stops on the left :
[tex]x=3\,cos(3\pi)-3\pi+5=-3\,-3\pi +5=2-3\pi\\y=sin(3\pi)=0[/tex]
