The least positive value x for which cot(x) = 0 is π/2.
The Correct Answer and Explanation is:
The least positive value of x for which cot(x) = 0 is π/2.
To understand this result, consider the definition of the cotangent function. The cotangent of an angle x is given by cot(x) = 1/tan(x). This means cot(x) is zero wherever tan(x) approaches infinity or is undefined.
The tangent function is defined as tan(x) = sin(x)/cos(x). This function is zero whenever sin(x) = 0, which occurs at integer multiples of π such as 0, π, and 2π. However, tangent is undefined when cos(x) = 0, which happens at odd multiples of π/2, including π/2, 3π/2, and 5π/2. Since cotangent is the reciprocal of tangent, cotangent is zero at precisely these points where tangent is undefined.
To determine the least positive x where cot(x) = 0, focus on the smallest positive odd multiple of π/2. The first instance where cosine equals zero is at π/2, making this the smallest solution.
A deeper analysis confirms this result by considering the unit circle. The cotangent function is represented by the ratio of the adjacent side to the opposite side in a right triangle. At π/2, the opposite side (sine value) reaches its maximum while the adjacent side (cosine value) becomes zero. Since cotangent is the reciprocal of tangent, this leads cotangent to equal zero.
In summary, the smallest positive value of x that satisfies cot(x) = 0 is π/2, as this is the first instance where cosine is zero and tangent is undefined, forcing cotangent to take the value zero.
