Consider the following: $x = 5 \cos(\theta), \quad y = 6 \sin(\theta), \quad -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$

(a) Eliminate the parameter to find a Cartesian equation of the curve.

Neetesh Kumar | January 3, 2025

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This is the solution to Math 1c
Assignment: 10.1 Question Number 13
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Step-by-step solution:

Step 1: Express $\cos(\theta)$ and $\sin(\theta)$ in terms of $x$ and $y$

From the first equation:

$x = 5 \cos(\theta)$, so $\cos(\theta) = \frac{x}{5}$.

From the second equation:

$y = 6 \sin(\theta)$, so $\sin(\theta) = \frac{y}{6}$.

Step 2: Use the Pythagorean identity

The Pythagorean identity is:

$\cos^2(\theta) + \sin^2(\theta) = 1$.

Substitute $\cos(\theta) = \frac{x}{5}$ and $\sin(\theta) = \frac{y}{6}$ into the identity:

$\left(\frac{x}{5}\right)^2 + \left(\frac{y}{6}\right)^2 = 1$.

Step 3: Simplify the equation

$\frac{x^2}{25} + \frac{y^2}{36} = 1$.

Final Cartesian Equation

The Cartesian equation of the curve is:

$\boxed{\frac{x^2}{25} + \frac{y^2}{36} = 1}$

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