Question: A disaster response team models the periodic power grid stability using the function $ p(t) = 9\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right) $. What is the amplitude and the phase shift $ \phi $ in the equivalent form $ R\cos\left(\frac{\pi}{12}t - \phi\right) $?

["Optimizing Disaster Response with Power Grid Stability: Analyzing the Periodic Function $ p(t) = 9\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right) $", "Understanding and predicting the stability of critical infrastructure like power grids during disaster events is a vital aspect of modern emergency planning. In modeling fluctuations in grid stability, engineers often use trigonometric functions to represent periodic behavior. A particularly common and insightful transformation involves converting a sum of cosine and sine terms into a single cosine function with an amplitude and phase shift. This approach simplifies analysis and enables timely response predictions.", "In this article, we explore how the period-averaging power grid stability function\n$$\np(t) = 9\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right)\n$$\ncan be rewritten in the form $ R\cos\left(\frac{\pi}{12}t - \phi\right) $, helping identify key stability patterns. We determine both the amplitude $ R $ and the phase shift $ \phi $, essential parameters for interpreting system resilience.", "---", "### Step 1: Expressing $ p(t) $ as a Single Cosine Function", "The general identity for combining sinusoidal functions is:\n$$\na\cos\ heta + b\sin\ heta = R\cos(\ heta - \phi)\n$$\nwhere\n$$\nR = \sqrt{a^2 + b^2}, \quad \ an\phi = \frac{b}{a}\n$$\nHere, $ a = 9 $, $ b = 12 $, and $ \ heta = \frac{\pi}{12}t $. Applying the formulas:", "1. Compute the amplitude:\n$$\nR = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15\n$$", "2. Compute the phase shift $ \phi $:\nUsing $ \ an\phi = \frac{12}{9} = \frac{4}{3} $, we find:\n$$\n\phi = \arctan\left(\frac{4}{3}\right)\n$$\nThis value represents a positive phase shift indicating the time lag (in hours, given frequency $ \frac{\pi}{12} $) before the peak instability occurs.", "---", "### Step 2: Interpreting Amplitude and Phase Shift for Grid Management", "- Amplitude $ R = 15 $:\n The function oscillates with a total variation of 30 (from $ -15 $ to $ 15 $), reflecting high dynamic fluctuations in grid stability under modeled stress conditions. A larger amplitude signals greater instability, prompting emergency teams to prioritize reinforcement or load-shedding protocols.", "- Phase shift $ \phi = \arctan\left(\frac{4}{3}\right) \approx 0.927 $ radians:\n This phase value (about $ 53^\circ $) indicates when maximum instability is anticipated after time $ t = \phi \cdot \frac{12}{\pi} \approx 3.54 $ hours. This timing insight improves response coordination during active disaster scenarios.", "---", "### Conclusion: Strengthening Disaster Response Through Precision Modeling", "By transforming $ p(t) $ into $ R\cos\left(\frac{\pi}{12}t - \phi\right) $, with $ R = 15 $ and $ \phi = \arctan\left(\frac{4}{3}\right) $, power grid analysts gain a compact yet powerful representation of stability dynamics. Such mathematical modeling supports faster, data-driven decisions during crises—ultimately enhancing the resilience of disaster response efforts.", "Understanding these parameters not only optimizes technical performance but also empowers teams to anticipate and mitigate risks before they escalate. For emergency planners and engineers, mastering trigonometric transformations of periodic functions like $ p(t) $ is a key step toward smarter infrastructure management.", "---", "Keywords: disaster response, power grid stability, $ p(t) = 9\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right) $, amplitude $ R $, phase shift $ \phi $, trigonometric modeling, frequency $ \frac{\pi}{12} $, emergency planning.", "---", "Note: The amplitude $ R = 15 $ and phase shift $ \phi = \arctan\left(\frac{4}{3}\right) $ fully characterize the equivalent form, enabling precise monitoring and predictive response in high-stakes operational environments."]