Final answer:
The curve generated by the parametric equations is a periodic wave-like shape, moving to the right as time t increases, with a period of 2π.
Step-by-step explanation:
The student is asking about the curve generated by the given parametric equations x = t + sin(t) and y = cos(t), within the range of −π ≤ t ≤ π. To identify the curve and indicate the direction in which the curve is traced as t increases, we can evaluate the equations at various values of t within the given range and construct a graph based on the calculated points.
First, recall that sin(t) and cos(t) are periodic functions with a period of 2π, which implies that the generated curve will most likely be periodic as well. For the y-coordinate, because y = cos(t), it oscillates between 1 and -1. The x-coordinates, given by x = t + sin(t), will oscillate as well but will have a constant increase due to the t term. Therefore, as t increases from −π to π, the x-coordinate will increase continuously while the y-coordinate follows a cosine oscillation. This will create a shape that looks like a wave moving to the right.
The curve will start at the point (−π, 1), since sin(−π) = 0 and cos(−π) = 1, and will end at the point (π, -1), since sin(π) = 0 and cos(π) = -1. Connecting these points with a smooth wave-like curve and marking the direction of increase, we get the desired graph. The direction of the curve (traced as t increases) will be shown with an arrow along the curve, moving from left to right.