Question

Solve the following exercise.sinx(t)=cost,y(t)=1−sint, for 0≤t≤2π. This is a circle offset from the origin.Based on the result, what is a parametric equation for a circle of radius 1 centered at (2, -1)? What is a parametric equation for a circle of radius 0.5 centered at (2, -1)?

Answer 1

The equations x(t) = 2 + cos(t) and y(t) = -1 + sin(t) represent a circle of radius 1 centered at (2, -1), while x(t) = 2 + 0.5 * cos(t) and y(t) = -1 + 0.5 * sin(t) represent a circle of radius 0.5 centered at the same point.

Step-by-step explanation:

The original equations sinx(t)=cost and y(t)=1−sint define a circle when plotted parametrically for 0≤t≤2π, since the functions 'sin' and 'cos' are orthogonal components of a circular motion. To find a parametric equation for a circle of radius 1 centered at (2, -1), we simply adjust the equations to translate the center of the circle:

For the circle of radius 1: x(t) = 2 + cos(t) and y(t) = -1 + sin(t).

Similarly, for the circle of radius 0.5 centered at (2, -1): x(t) = 2 + 0.5 * cos(t) and y(t) = -1 + 0.5 * sin(t).