7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

Verified step by step guidance

1. Identify the given parametric equations: \(x = 8\cos t + 1\) and \(y = 8\sin t + 2\) with the parameter \(t\) in the interval \$0 \leq t \leq 2\pi$.
2. Recall the Pythagorean identity: \(\cos^2 t + \sin^2 t = 1\). This will help us eliminate the parameter \(t\) by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\).
3. Isolate \(\cos t\) and \(\sin t\) from the parametric equations: \(\cos t = \frac{x - 1}{8}\) and \(\sin t = \frac{y - 2}{8}\).
4. Substitute these expressions into the Pythagorean identity to get an equation involving only \(x\) and \(y\): \(\left(\frac{x - 1}{8}\right)^2 + \left(\frac{y - 2}{8}\right)^2 = 1\).
5. Simplify the equation by multiplying both sides by \$64\( (since \)8^2 = 64\() to obtain the Cartesian equation of the curve without the parameter \)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 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.

Tangent Lines to Parametric Curves

The tangent line to a parametric curve at a given parameter value t is found by computing derivatives dx/dt and dy/dt, then using dy/dx = (dy/dt)/(dx/dt). This slope helps write the equation of the tangent line at the specified point on the curve.