Final answer:
To find the equation of the circle of curvature at a specific point on the curve r(t) = 4ti + sin(5t)j, one must compute the radius of curvature and find the center of the circle, which lies at a distance R along the normal line to the curve at the point. Without the derivatives of the curve, it's not possible to provide the exact formula.
Step-by-step explanation:
The goal is to find the equation for the circle of curvature of the curve r(t) = 4ti + sin(5t)j at the point (2π,1). The curve given represents the graph of y = sin(5x/4) in the xy-plane. To find the circle of curvature, we need to find the radius of curvature (R) at the specified point and then determine the center of the circle of curvature, ensuring the circle is tangent to the curve at the point of interest.
To find the radius of curvature, we use the formula R = |ς(t)|/(|κ(t)|), where ς(t) is the radius vector and κ(t) is the curvature of the curve. Unfortunately, the question didn't provide enough information to directly compute the radius of curvature, such as the derivatives of r(t). Therefore, I cannot confidently provide an accurate formula for the circle of curvature without additional information on the derivatives of the curve. However, once R is known, the equation of the circle can be written as (x - h)^2 + (y - k)^2 = R^2, where (h, k) are the coordinates of the center of the circle, located at a distance R from the point of curve along the normal line.