56. \( \cot x=-1 \)

Did you know that the equation \( \cot x = -1 \) leads us to some interesting angles? Since cotangent is the reciprocal of tangent, this means \( \tan x = -1 \). The angles where tangent equals -1 occur in the second (where sine is positive, cosine is negative) and fourth quadrants (where sine is negative, cosine is positive). Specifically, these angles are \( x = \frac{3\pi}{4} + k\pi \) for any integer \( k \), which indicates the periodic nature of the function! In real-world applications, angles where cotangent, tangent, or other trigonometric functions equal specific values can be seen in navigation, physics, and engineering. For instance, they can help determine the slope of an incline in construction. Understanding these trigonometric identities allows architects and engineers to design structures with precise angles, ensuring they are both safe and aesthetically pleasing!