Final answer:
To find the tangent line to the curve at t = π, determine the point on the curve (-π, 0), calculate the slope of the tangent at that point to be -1, and then use the point-slope form to write the equation of the tangent as y = -x - π.
Step-by-step explanation:
To find an equation of the tangent to the curve at the parameter t = π, we first find the coordinates of the point on the curve. For the given parameterized equations x = tcos(t) and y = tsin(t), at t = π, we get x = πcos(π) and y = πsin(π). Plugging in the value of π, we get the point (x, y) = (-π, 0).
Next, we need to find the slope of the tangent line at this point. We do this by finding the first derivatives dx/dt and dy/dt and then evaluating the slope, v, at t = π, which gives us the rate of change of y with respect to x. The derivative of x with respect to t is -tsin(t) + cos(t) and of y with respect to t is tcos(t) + sin(t). At t = π, the derivatives become π + 0 and -π + 0 respectively, thus v = dy/dx = (tcos(t) + sin(t))/(-tsin(t) + cos(t)) = -1 at t = π.
The equation of the tangent line can then be constructed using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the point on the curve and m is the slope at that point. Plugging in our values, the equation of the tangent line at t = π is: y - 0 = -1(x + π) or y = -x - π.