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Home » Use the double-angle identity for cosine: - AIKO, infinite ways to autonomy.

Use the double-angle identity for cosine: - AIKO, infinite ways to autonomy.

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: A robotics engineer is optimizing a control system where the error function is defined as $ E = a(a + b) + b(a + b) $, and $ a + b = 12 $. What is the value of $ E $?

📌 Solution: Given $ E = a(a + b) + b(a + b) $, factor out $ (a + b) $:

📌 Substitute $ a + b = 7 $, $ ab = 10 $:

📌 Since the sum $x + y + z$ modulo 7 is uniformly distributed over the 7 possible residues (0 through 6), and the cubes are symmetric and large, we expect the number of triples with each residue class to be approximately equal.

📌 So $k = 8m$, and $n = 125(8m) + 38 = 1000m + 38$. The smallest positive solution is $n = 38$.

📌 Question: A materials scientist is modeling the crystal lattice of a new self-healing polymer and finds that the number of atoms in a unit cell satisfies a number-theoretic condition: it is a three-digit number that is one less than a multiple of 11 and one more than a multiple of 13. What is the smallest such number?

📌 Solution: We seek the smallest three-digit number $n$ such that:

📌 Use the Chinese Remainder Theorem. Let $n = 13k + 1$. Substitute into the first congruence:

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