Final answer:
To find the equation of the line normal to the curve at a given point, we need to find the derivative of the curve at that point using the chain rule. Then, we substitute the coordinates of the given point into the equation to find the equation of the line.
Step-by-step explanation:
To find the equation of the line normal to the curve at a given point, we need to find the derivative of the curve at that point. Given the equation sin(y) = x cos(2y), we first differentiate both sides with respect to x using the chain rule:
cos(y) * dy/dx = cos(2y) - 2x sin(2y) * dy/dx.
Next, we solve for dy/dx by isolating it on one side of the equation:
dy/dx = (cos(2y))/(cos(y) - 2x sin(2y))
This is the derivative of the curve at the given point. The equation of the line normal to the curve at the given point can be found by finding the negative reciprocal of the derivative and substituting the coordinates of the given point into the equation.