Final answer:
To find the equation of the tangent line to the curve at a given point, we need to find the slope of the curve at that point. By finding the values of theta for the given x-coordinates and then calculating dy/dx using the chain rule, we can determine the slope of the tangent line at each point.
Step-by-step explanation:
To find the equation of the tangent line to the curve at a given point, we need to find the slope of the curve at that point. In this case, the curve is defined parametrically by the equations x = 2 - 3cos(theta) and y = 3 + 2sin(theta). We are asked to find the equation of the tangent line at (-1,3) and (2,5).
First, we need to find the values of theta for which x = -1 and x = 2. By solving the equations, we find that theta = 0 and theta = pi. Plugging these values into the equations for y, we get y = 3 and y = 5.
Now, we have the coordinates of two points on the curve (-1,3) and (2,5) as well as the values of theta for those points. The slope of the tangent line at a given point is equal to the derivative of y with respect to x at that point. To find it, we can calculate dy/dx for each point using the chain rule and plug in the values of theta.