Question

the ellipse \\(\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1\\) can be drawn with parametric equations where \\(x(t)\\) is written in the form \\(x(t) = r\cos(t)\\) with \\(r = \square\\) and \\(y(t) = \square\\) question help: \\(\boxed{\text{video}}\\)

Explanation:

Step1: Recall Ellipse Parametric Form

The standard ellipse equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Its parametric equations are $x = a\cos(t)$, $y = b\sin(t)$.

Step2: Identify \( r \) (i.e., \( a \))

Given ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, compare with standard form. For $x(t)=r\cos(t)$, $r = a = 3$.

Step3: Find \( y(t) \)

From parametric form, $y(t)=b\sin(t)$. Here, $b = 2$, so $y(t)=2\sin(t)$.

Answer:

For \( r \): \( 3 \)
For \( y(t) \): \( 2\sin(t) \)