Question

A pair of parametric equations is given. x = tan(t), y = cot(t), 0 < t < pi/2 Find a rectangular-coordinate equation for the curve by eliminating the parameter. __________ , where x > _____ and y > ______

Answer 1

To eliminate the parameter t and find a rectangular-coordinate equation for the curve defined by x = tan(t) and y = cot(t), t can be expressed as t = arctan(x) and cot(t) can be expressed as cot(t) = 1/tan(t) = 1/(arctan(x)). Substituting these values into the given equations, we get x = tan(arctan(x)) = x and y = cot(arctan(x)) = 1/(arctan(x)). Therefore, the rectangular-coordinate equation for the curve is y = 1/(arctan(x)), where x > 0 and y > 0.

Step-by-step explanation:

To eliminate the parameter t and find a rectangular-coordinate equation for the curve defined by x = tan(t) and y = cot(t), we need to express tan(t) and cot(t) in terms of x and y. The reciprocal identity of tangent and cotangent can be used to solve for t in terms of x and y.

t can be expressed as t = arctan(x) and cot(t) can be expressed as cot(t) = 1/tan(t) = 1/(arctan(x)).

Substituting these values into the given equations, we get x = tan(arctan(x)) = x and y = cot(arctan(x)) = 1/(arctan(x)). Therefore, the rectangular-coordinate equation for the curve is y = 1/(arctan(x)), where x > 0 and y > 0.