SOLUTION: Hi i'm not to sure if this is in the right section but hope so. I'm currently having problems with the following question. Given a parametric curve x = 2sint, y=sin(2t) where t

Question 863222: Hi i'm not to sure if this is in the right section but hope so. I'm currently having problems with the following question.
Given a parametric curve x = 2sint, y=sin(2t) where t takes all values in R (real numbers)
(a) Find an implicit equation of this curve by eliminating t,
(b) Sketch this parametric curve in the xy-plane.
I'm more confused about a , I don't expect anyone to do (b) perhaps just inform me if my thoughts are correct.
Do i set it so that 0 = arcsin(x/2) - (arcsin(y))/2 --> from setting t=t OR somewhere along the lines of x + y = 2sint + sin(2t) and it's somehow a circle.. Or am I just completely off.. Thanks to anyone who responds !!
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
x = 2sin(t)
y = sin(2t)

as far as i can tell, you eliminate t by solving for t in one of the equations and then using that value of t in the other equation.

your 2 equations are:

x = 2sin(t)
y = sin(2t)

you solve for t in the first equation.
start with:
x = 2sin(t)
divide both sides by 2 to get:
x/2 = sin(t)
this is true if and only if arcsin(x/2) = t
replace t with arcsin(x/2) in the second equation to get:
y = sin(2*arcsin(x/2))
your equations are now in terms of x and f(x).
the equation you can graph is y = sin(2*arcsin(x/2))
finding the domain of this equation can be a challenge.
since the sine of an angle is between -1 and 1, this means that the arcsine of a number is only valid when the number is between -1 and 1.
since the number is x/2, this means that x has to be between -2 and 2 only.
in order for x to be between -2 and 2 only, this means that t has to be between -pi/2 and pi/2 only.
the graph of your equation is shown below:
$$$

the same graph with associated data table created in Excel is shown below:
$$$

a decent reference on parametric equations is shown below:
http://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx

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