Find two different parametric descriptions for the circle of radius 4 centered at (-3,2).
I can only think of one
(x,y) = ( -3+ 4sint, 2+ 4cost )
Well for that author it is not incorrect. But as far as I am concerned it is.
The cosine is a function. It's symbol is $\displaystyle \cos$.
Like all functions we write $\displaystyle \cos(t)$ as its value at t.
That is definitely incorrect period.
The $\displaystyle \cos(t)$ is associated with the x-coordinate and $\displaystyle \sin(t)$ is associated with the y-coordinate.
So in ordered pair notation the first circle would be:
$\displaystyle \left( {4\cos (t) - 3,4\sin (t) + 2} \right)$.
Please excuse my ignorance, but I don't understand why the equation (x,y) = (5sin (12t) , 6- 5cos(12t)) is incorrect.
Can't I look at x as a function of values of 5sin(12t) and y as a function of the values of 6-5cos(12t)
If I plug in t=1
x= 1.039558454 y= 1.11
t=2
x = 2.033683215 y= 1.432273
If I keep on going for different values of t and connect all the dots
I will end up with the equation x squared + ( y-6 ) squared = 25
Like plato already said, $\displaystyle \cos(t)$ is associated with the x-coordinate and $\displaystyle \sin(t)$ with the y-coordinate.
Remark:
If you don't see that you can draw a circle with a chosen radius r centered at (0,0), take a point P on the circle and project this on the x-axis and y-axis. With the proposition of Phytagoras you'll find the coordinates $\displaystyle r\cos(t)$ and $\displaystyle r \sin(t)$ so you get $\displaystyle P(r \cos(t), r \sin (t))$ and not $\displaystyle P(r \sin(t), r \cos (t))$
For example, for simplicity take a point on the circumference in the 1st quadrant.
Draw a right-angled triangle from that point to the point on the x-axis directly below it and the circle centre.
Now if we label the angle in the triangle where it touches the circumference as "t",
then we obtain an alternative parametric representation.
I disagree. There is nothing wrong with the parametrisation $\displaystyle (x,y) = ( 5\sin(12t) , 6-5\cos(12t))$ (for a circle centred at (0,6) with radius 5). It is unusual to have sin associated with the x-coordinate and cos with the y-coordinate. But it is certainly not incorrect. In fact, the question asks for alternative parametrisations, and that would be a perfectly reasonable way of obtaining one.