7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6

Verified step by step guidance

Identify the given parametric equations: \(x = 4\sin(2t)\) and \(y = 3\cos(2t)\), with the parameter \(t\) in the interval \(0 \leq t \leq \pi\).

Recall that to eliminate the parameter \(t\), we want to express \(y\) directly in terms of \(x\) without involving \(t\).

Use the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). Since both \(x\) and \(y\) involve \(\sin(2t)\) and \(\cos(2t)\), rewrite each in terms of \(\sin(2t)\) and \(\cos(2t)\):

\[ \sin(2t) = \frac{x}{4} \quad \text{and} \quad \cos(2t) = \frac{y}{3} \]

Substitute these into the Pythagorean identity to get an equation relating \(x\) and \(y\) without \(t\): \[ \left(\frac{x}{4}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Tangent Lines to Parametric Curves

The tangent line to a parametric curve at a given parameter value t is found by computing dy/dx as (dy/dt) / (dx/dt). This derivative gives the slope of the tangent line, which can then be used with the point coordinates to write the equation of the tangent line.

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