In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 2 + 3 cos t, y = 4 + 2 sin t; t = π

Verified step by step guidance

1. Identify the parametric equations given: \(x = 2 + 3 \cos t\) and \(y = 4 + 2 \sin t\).
2. Substitute the given value of the parameter \(t = \pi\) into the equation for \(x\): calculate \(x = 2 + 3 \cos \pi\).
3. Recall the value of \(\cos \pi\), which is \(-1\), and use it to simplify the expression for \(x\).
4. Substitute the same value \(t = \pi\) into the equation for \(y\): calculate \(y = 4 + 2 \sin \pi\).
5. Recall the value of \(\sin \pi\), which is \(0\), and use it to simplify the expression for \(y\). The coordinates of the point are then \((x, y)\).

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 ellipses or circles.

Evaluating Trigonometric Functions at Specific Angles

To find coordinates for a given parameter t, substitute t into the trigonometric functions (cos t and sin t). Knowing exact values of sine and cosine at common angles like π is essential for accurate calculation of points on the curve.

Coordinate Calculation from Parametric Form

Once the parameter value is substituted, calculate x and y by evaluating the expressions. This process converts the parametric form into a specific point (x, y) on the plane, representing the location on the curve at that parameter.