The tangent to the curve with equation y = tan(2x) at the point where x = π8 meets they-axis at the point A. Find the distance OA, where O is the origin

Step 1: Find the derivative of y = tan(2x)

The derivative of y = tan(2x) is y' = 2sec^2(2x) using the chain rule.

Step 2: Evaluate the derivative at x = π/8

Substitute x = π/8 into the derivative to find the slope of the tangent line at that point.

y'(π/8) = 2sec^2(2π/8) = 2sec^2(π/4) = 22 = 4

So, the slope of the tangent line at x = π/8 is 4.

Step 3: Find the equation of the tangent line

The equation of a line is y = mx + b, where m is the slope and b is the y-intercept. We know that the slope m is 4, and the line passes through the point (π/8, tan(2*π/8)) = (π/8, 1). So, we can substitute these values into the equation to find b.

1 = 4*π/8 + b b = 1 - π/2

So, the equation of the tangent line is y = 4x + 1 - π/2.

Step 4: Find the y-intercept of the tangent line

The y-intercept of the line is the value of y when x = 0. So, substitute x = 0 into the equation of the line to find the y-intercept.

y = 4*0 + 1 - π/2 = 1 - π/2

So, the tangent line meets the y-axis at the point A(0, 1 - π/2).

Step 5: Find the distance OA

The distance OA is the distance from the origin O(0, 0) to the point A(0, 1 - π/2). This is simply the absolute value of the y-coordinate of A, which is |1 - π/2|.

So, the distance OA is |1 - π/2|.