Derive Theoretical Result

Prove geometric theorems using Euclidean axiomatic methods, coordinate geometry, or vector methods with rigorous step-by-step logical structure. Covers direct proof, proof by contradiction, coordinate proofs, vector proofs, and handling of special cases and degenerate configurations. Use when given a geometric statement to prove, verifying a conjecture, establishing a lemma, converting geometric intuition into a rigorous proof, or comparing the effectiveness of different proof methods.

Solve trigonometric equations and triangle problems systematically using identities, law of sines/cosines, inverse functions, and unit circle analysis. Covers equation solving, triangle resolution, identity verification, and applied trigonometric modeling. Use when solving trigonometric equations for unknown angles, resolving triangles from partial information (SSS, SAS, ASA), verifying identities, or applying trigonometry to real-world problems in surveying, physics, or engineering.

Analyze prime numbers using primality tests, factorization algorithms, prime distribution analysis, and sieve methods. Covers trial division, Miller-Rabin, Sieve of Eratosthenes, and the Prime Number Theorem. Use when determining whether an integer is prime or composite, finding prime factorizations, counting or listing primes up to a bound, or investigating prime properties within a number-theoretic proof or computation.

Solve problems involving changing magnetic flux using Faraday's law, Lenz's law, motional EMF, mutual and self-inductance, and RL circuit transients. Use when computing induced EMF from time-varying B-fields or moving conductors, determining current direction via Lenz's law, analyzing inductance and energy storage in magnetic fields, or solving RL circuit differential equations for switching transients.

Quantum physics simulation library for open quantum systems. Use when studying master equations, Lindblad dynamics, decoherence, quantum optics, or cavity QED. Best for physics research, open system dynamics, and educational simulations. NOT for circuit-based quantum computing—use qiskit, cirq, or pennylane for quantum algorithms and hardware execution.

Solve Diophantine equations (integer-only solutions) including linear, quadratic, and Pell equations. Covers the extended Euclidean algorithm, descent methods, and existence proofs. Use when finding all integer solutions to ax + by = c, solving Pell's equation, generating Pythagorean triples, proving no integer solutions exist via modular constraints, or finding the fundamental solution from which all others are generated.