Final answer:
To find the equation of the tangent line at the point (π/6, 3) for the curve y = 6sin(x), we first determine the derivative, which is y' = 6cos(x), and then evaluate it at x = π/6 to get the slope m = 3√3. Using the point-slope equation, we have y - 3 = 3√3(x - π/6), which can be rearranged to y = mx + b to solve for the y-intercept (b).
Step-by-step explanation:
To find the equation of the tangent line to the curve y = 6sin(x) at the point (π/6, 3), first, you need to calculate the derivative of y with respect to x, which represents the slope of the tangent line at any point on the curve. The derivative of 6sin(x) is 6cos(x). Next, evaluate the derivative at the point x = π/6 to find the slope of the tangent line at that point.
The slope (m) at x = π/6 is 6cos(π/6) = 6 ∙ (√3/2) = 3√3.
Next, using the point-slope form of the equation y - y1 = m(x - x1), you can plug in the point we have and the slope we found:
y - 3 = 3√3(x - π/6).
To find the y-intercept (b), rearrange the equation to y = mx + b and solve for b when x is 0.