Parametric Equations

[1141]

I've started doing parametric equations, and I need help with a certain question.

The question is:

"A curve is given by x = 5cost , y = 2sint for 0? t < 2?. Find the value of t at the point (-2 1/2; ?3)."

I've got two equations so far:

x = 5cost
-2 1/2 = 5cost

and

y = 2sint
?3 = 2sint


I do not know what to do next. Do I have to do a simultaneous equation or something?

 

[soroban]

Hello, 1141!

Your two equations are correct . . . Keep going!

\(\displaystyle \text{A curve is given by: }\;\begin{Bmatrix}x &=& 5\cos t \\y &=& 2\sin t\end{Bmatrix}\;\text{ for }0\,\le\, t\,\le\,2\pi\)

\(\displaystyle \text{Find the value of }t\text{ at the point }\left(\text{-}\tfrac{5}{2},\; \sqrt{3}\right)\)

\(\displaystyle \text{We have: }\;x \:=\: 5\cos t \quad\Rightarrow\quad \text{-}\tfrac{5}{2} \:=\:5\cos t\)

. . \(\displaystyle \text{Then: }\;\cos t \:=\:\text{-}\tfrac{1}{2} \quad\Rightarrow\quad t \:=\:\tfrac{\pi}{3}\text{ or }\tfrac{5\pi}{3}\)


\(\displaystyle \text{We have: }\;y \:=\: 2\sin t \quad\Rightarrow\quad \sqrt{3} \:=\:2\sin\)

. . \(\displaystyle \text{Then: }\;\sin t \:=\:\tfrac{\sqrt{3}}{2} \quad\Rightarrow\quad t \:=\:\tfrac{\pi}{3}\text{ or }\tfrac{2\pi}{3}\)


\(\displaystyle \text{Therefore: }\;t \:=\:\frac{\pi}{3}\)