Question

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. (If an answer does not exist, enter DNE.)

x = cos(t), y = 8 sin(t), t = π/2

slope dy/dx = ....

tangent line equation .....

Answer 1

The slope of the tangent line at t = π/2 for the parametric equations x = cos(t) and y = 8 sin(t) is 0. The equation of the tangent line at this point is y = 8.

Step-by-step explanation:

To determine the slope of the tangent line to the curve given by x = cos(t) and y = 8 sin(t) at a specific value of the parameter t, we need to find the derivatives of x and y with respect to t, and then use them to compute dy/dx. This will give us the slope of the tangent at that point.

At t = π/2, we have:

• x'(t) = -sin(t)
• y'(t) = 8 cos(t)

Now we find the derivatives at t = π/2:

• x'(π/2) = -sin(π/2) = -1
• y'(π/2) = 8 cos(π/2) = 8(0) = 0

The slope dy/dx is the ratio of y' to x', hence at t = π/2, dy/dx = 0 / -1 = 0. This indicates that the tangent line is horizontal.

Now to find the equation of the tangent line, we substitute t = π/2 into the parametric equations to get the point of tangency: x = cos(π/2) = 0, y = 8 sin(π/2) = 8. The equation of a horizontal line passing through (0,8) is y = 8.