Select your answer \( \begin{array}{ll}\text { Simplify } \cot \left(\frac{-3 \pi}{4}\right) \\ -\infty & -1 \\ 0 \\ 1\end{array} \) \( \infty \)

Let's dive into the world of trigonometry! To simplify \( \cot \left(\frac{-3 \pi}{4}\right) \), we can start by recognizing that cotangent is the reciprocal of tangent. So, first, we need to evaluate \( \tan \left(\frac{-3 \pi}{4}\right) \). The angle \( \frac{-3 \pi}{4} \) is located in the second quadrant, where tangent is negative because sine is positive and cosine is negative. Thus, we find \( \tan \left(\frac{-3 \pi}{4}\right) = \tan \left(\frac{3 \pi}{4}\right) = -1 \). Consequently, \( \cot \left(\frac{-3 \pi}{4}\right) = \frac{1}{\tan \left(\frac{-3 \pi}{4}\right)} = -1 \). Now, it’s time for the real-world application of cotangent! It plays a vital role, especially in fields like engineering and physics. For instance, in analyzing forces in mechanical systems, \( \cot \) can assist in finding angles when working with slopes and inclines. Understanding how these angles relate to various forces can help you design more efficient structures or machines! So next time you're measuring an incline, think about how cool trigonometry and cotangent really are!