For the parametric curve x = cos t, y = sin t, what is dy/dx in terms of t?

For the parametric curve x = cos t, y = sin t, what is dy/dx in terms of t?

• A. -cot t
• B. tan t
• C. cot t
• D. -tan t

Answer: (A)

The slope of a parametric curve is found by dy/dx = (dy/dt) / (dx/dt). For x = cos t and y = sin t, we have dy/dt = cos t and dx/dt = -sin t. Therefore dy/dx = (cos t)/(-sin t) = -cot t. This is defined when sin t ≠ 0; at t where sin t = 0 the slope is not defined (vertical tangent), which matches cot t being undefined there. So the derivative in terms of t is -cot t.

The slope of a parametric curve is found by dy/dx = (dy/dt) / (dx/dt). For x = cos t and y = sin t, we have dy/dt = cos t and dx/dt = -sin t. Therefore dy/dx = (cos t)/(-sin t) = -cot t. This is defined when sin t ≠ 0; at t where sin t = 0 the slope is not defined (vertical tangent), which matches cot t being undefined there. So the derivative in terms of t is -cot t.