To find the maximum intensity of the function \( I(t) = 5\sin(t) + 3\cos(t) \), we can express it in the form \( R\sin(t + \alpha) \).
["# Finding the Maximum Intensity of the Function ( I(t) = 5\sin(t) + 3\cos(t) )", "When analyzing periodic functions, one key question often arises: what is the maximum intensity of the function? For the function\n[ I(t) = 5\sin(t) + 3\cos(t), ]\na powerful mathematical approach is expressing it in the form ( R\sin(t + \alpha) ). This transformation simplifies determining the maximum value, making it easier to analyze amplitude and behavior.", "---", "### Why Rewrite as ( R\sin(t + \alpha) )?", "The expression ( R\sin(t + \alpha) ) represents a single sinusoidal wave with amplitude ( R ), phase shift ( \alpha ), and frequency matching the original function. Since the maximum value of any sine function is 1, the maximum value of ( R\sin(t + \alpha) ) is simply ( R ). Thus, finding ( R ) gives the maximum intensity of ( I(t) ).", "---", "### Deriving the Amplitude ( R )", "We begin by applying the standard trigonometric identity for combining sine and cosine terms:", "[\na\sin(t) + b\cos(t) = R\sin(t + \alpha),\n]\nwhere\n[\nR = \sqrt{a^2 + b^2}, \quad \ ext{and} \quad \ an(\alpha) = \frac{b}{a}.\n]", "For our function ( I(t) = 5\sin(t) + 3\cos(t) ):\n- ( a = 5 )\n- ( b = 3 )", "So, compute ( R ):", "[\nR = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}\n]", "---", "### Maximum Intensity of ( I(t) )", "Since ( I(t) = R\sin(t + \alpha) = \sqrt{34}\sin(t + \alpha) ), and the maximum value of ( \sin ) is ( 1 ), the maximum intensity is:", "[\n\ ext{Max } I(t) = \sqrt{34}\n]", "---", "### Conclusion", "By rewriting ( I(t) = 5\sin(t) + 3\cos(t) ) in the form ( R\sin(t + \alpha) ), we find that the maximum intensity is ( \sqrt{34} ). This insight is valuable in fields like signal processing, physics, and engineering where determining peak values is essential. No matter how the sine and cosine components shift in time, their combined amplitude—the maximum output value—depends only on ( R = \sqrt{a^2 + b^2} ).", "---", "### Key Takeaways", "- Expressing ( 5\sin(t) + 3\cos(t) ) as ( R\sin(t + \alpha) ) reveals the maximum intensity directly.\n- The amplitude is ( R = \sqrt{5^2 + 3^2} = \sqrt{34} ).\n- Thus, the maximum value of ( I(t) ) is ( \boxed{\sqrt{34}} ).", "Use this method whenever you analyze wave-like functions—rewriting in amplitude-phase form simplifies finding maxima and enhances understanding of the function’s behavior."]
