10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)

Verified step by step guidance

1. Recognize that the given parametric equations are \(x = 3\cos(-t)\) and \(y = 3\sin(-t) - 1\) with the parameter \(t\) in the interval \(0 \leq t \leq \pi\).
2. Recall the trigonometric identities for cosine and sine of negative angles: \(\cos(-t) = \cos t\) and \(\sin(-t) = -\sin t\). Use these to rewrite the parametric equations as \(x = 3\cos t\) and \(y = -3\sin t - 1\).
3. Isolate the trigonometric functions from the parametric equations: \(\cos t = \frac{x}{3}\) and \(\sin t = -\frac{y + 1}{3}\).
4. Use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to eliminate the parameter \(t\). Substitute the expressions for \(\sin t\) and \(\cos t\) into this identity:
5. \[\left(\frac{x}{3}\right)^2 + \left(-\frac{y + 1}{3}\right)^2 = 1.\]
6. Simplify the equation to obtain a relation purely in terms of \(x\) and \(y\), which represents the Cartesian equation of the curve without the parameter \(t\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.

Trigonometric Identities

Trigonometric identities, such as sin(-t) = -sin(t) and cos(-t) = cos(t), are essential for simplifying parametric equations involving sine and cosine. These identities help transform and combine expressions to eliminate the parameter and find the Cartesian equation.