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 = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, often denoted as 't'. In this case, x and y are defined in terms of t, allowing for the representation of complex shapes and motions in a two-dimensional plane. Understanding how to evaluate these equations at specific values of t is crucial for finding points on the curve.

Trigonometric Functions

The equations provided involve cosine and sine functions, which are fundamental trigonometric functions. These functions relate angles to the ratios of sides in right triangles and are periodic, meaning they repeat values at regular intervals. Knowing the values of sine and cosine at key angles, such as π/2, is essential for calculating the coordinates accurately.

Coordinate System

A coordinate system allows us to locate points in a plane using pairs of numbers (x, y). In this context, the x-coordinate and y-coordinate are derived from the parametric equations. Understanding how to interpret these coordinates in relation to the graph of the curve is important for visualizing the point's position on the plane.