Final answer:
To find the equation of the tangent line to the curve y = sin(5x)cos(6x) at a specific point, we need to find the derivative of the curve and evaluate it at the given point. Once we have the slope, we can use the point-slope form of a line to find the equation of the tangent line.
Step-by-step explanation:
To find the equation of the tangent line to the curve y = sin(5x)cos(6x) at a specific point, we need to find the derivative of the curve. Let's start by finding the derivative of y with respect to x:
y' = (d/dx)[sin(5x)cos(6x)]
Using the product rule and the chain rule, we can simplify the derivative to:
y' = 5cos(5x)cos(6x) - 6sin(5x)sin(6x)
Now, evaluate y' at the given point to find the slope of the tangent line. Substitute x = 25 into the equation for y' to get the slope:
y'(25) = 5cos(125)cos(150) - 6sin(125)sin(150)
Simplify the expression to find the slope of the tangent line at x = 25. Once you have the slope, you can use the point-slope form of a line, y - y₁ = m(x - x₁), to find the equation of the tangent line.