Final answer:
To find the equation of the tangent line to the curve at the point (pi/6, 4), calculate the derivative (8cosx) and evaluate at x = pi/6 to get the slope. Using point-slope form, we determine the y-intercept when x=0. The final equation of the tangent is y = 4\sqrt{3}x - 2\sqrt{3}pi + 4, with slope m = 4\sqrt{3} and y-intercept b = -2\sqrt{3}pi + 4.
Step-by-step explanation:
Finding the Slope and Y-Intercept of the Tangent Line
To find the equation of the tangent line to the curve y=8sinx at the point (pi/6,4), we first need to determine the slope of the tangent line, which is the derivative of y with respect to x evaluated at the point x = pi/6. The derivative of 8sinx with respect to x is 8cosx. Evaluating at x = pi/6, we get the slope m = 8cos(pi/6) = 8(\sqrt{3}/2) = 4\sqrt{3}.
Next, we use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line (in this case, (pi/6, 4)) and m is the slope. Plugging in the values, we have y - 4 = 4\sqrt{3}(x - pi/6). To find the y-intercept b, we solve for y when x=0: y = 4\sqrt{3}(0 - pi/6) + 4 = -2\sqrt{3}pi + 4.
The equation of the tangent line in slope-intercept form y = mx + b is then y = 4\sqrt{3}x - 2\sqrt{3}pi + 4, where m = 4\sqrt{3} and b = -2\sqrt{3}pi + 4.