Final answer:
To find the equation of the line tangent to y=f(x) when x=π/3, find the derivative of f(x), evaluate it at x=π/3, and use the point-slope form of a line.
Step-by-step explanation:
To find the equation of the line tangent to y=f(x) when x=π/3, we need to find the derivative of f(x) and evaluate it at x=π/3.
The derivative of f(x) = 4cos(x) is f'(x) = -4sin(x).
Evaluating f'(x) at x=π/3, we get f'(π/3) = -4sin(π/3) = -4√3/2.
So, the slope of the tangent line when x=π/3 is -4√3/2.
Since the line is tangent, it will pass through the point (x,y) = (π/3, 4cos(π/3)).
Using the point-slope form of a line, we can write the equation of the tangent line as y - 4cos(π/3) = (-4√3/2)(x - π/3).
Finally, simplifying the equation, we get y = -4√3/2(x - π/3) + 4cos(π/3).