Evaluate dy/dx at t = π/4 for x = cos t, y = sin t.

Evaluate dy/dx at t = π/4 for x = cos t, y = sin t.

• A. -2
• B. -1/2
• C. 0
• D. -1

Answer: (D)

When a curve is given parametrically, the slope dy/dx is found by dividing the rate of change of y with respect to t by the rate of change of x with respect to t: dy/dx = (dy/dt) / (dx/dt). Here, dx/dt = -sin t and dy/dt = cos t, so dy/dx = cos t / (-sin t) = -cot t. At t = π/4, sin t and cos t are both √2/2, so cot t = 1 and dy/dx = -1. The corresponding point is (√2/2, √2/2) on the unit circle, and the slope there is -1.

When a curve is given parametrically, the slope dy/dx is found by dividing the rate of change of y with respect to t by the rate of change of x with respect to t: dy/dx = (dy/dt) / (dx/dt).

Here, dx/dt = -sin t and dy/dt = cos t, so dy/dx = cos t / (-sin t) = -cot t. At t = π/4, sin t and cos t are both √2/2, so cot t = 1 and dy/dx = -1. The corresponding point is (√2/2, √2/2) on the unit circle, and the slope there is -1.