3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.
x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

Verified step by step guidance

1. Identify the parametric equations given: \(x = \sin t - 3\) and \(y = \cos t + 6\), with the parameter \(t\) ranging from \$0\( to \)\pi$.
2. Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This will help eliminate the parameter \(t\) by expressing \(x\) and \(y\) in terms of sine and cosine.
3. Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\): from \(x = \sin t - 3\), we get \(\sin t = x + 3\); from \(y = \cos t + 6\), we get \(\cos t = y - 6\).
4. Substitute these expressions into the Pythagorean identity: \((x + 3)^2 + (y - 6)^2 = 1\). This is the Cartesian equation of the curve.
5. Interpret the equation: it represents a circle centered at \((-3, 6)\) with radius \$1\(. The parameter \)t\( from \)0\( to \)\pi\( corresponds to the upper half of the circle, and the positive orientation follows the direction of increasing \)t\( from \)0\( to \)\pi$.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Understanding how x and y depend on t allows us to describe the curve's path and behavior over the given interval.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single equation in x and y. This process often uses algebraic manipulation and trigonometric identities to remove t, yielding a Cartesian equation that describes the curve.

Geometric Description and Orientation

After finding the Cartesian form, interpreting the curve geometrically involves identifying its shape (e.g., circle, ellipse) and position. Positive orientation refers to the direction in which the curve is traced as the parameter increases, important for understanding motion along the curve.