Preamble
novaTopFlex returns to a mathematic conquest of the journey to the top, the gateway to pure mathematic joy and excellence, and all the way to the purest forms of excitement! The novaTop community shall anticipate the highest performance levels possible in every respect in terms of what is considered ideal, desirable, and acceptable, with very high standards to ensure accuracy, fairness, and correctness. Starting with phase one, where a calculator is allowed, recommended, and encouraged, but going into phases two and three, as novaTop approaches the state and national conquest stages, it gets very important to be prepared for every possible next step.
Curious Facts about the Journey
Individuals and groups may understand trigonometry and related mathematics topics at a basic level, but the common people may not know, for instance, that adding the arctangent of one-third to the arctangent of one-seventh results in the arctangent of one-half. Additional quirks also include the identities which occur when cosines of angles are multiplied by cosines of sixty degrees minus the respective angles followed by cosines of sixty degrees plus the respective angles. With both sine and cosine relationships, the identities are identified as triple-angle identities thence divided by four.
Additional formulae can be derived from existing trigonometric properties, including from a baseline understanding of the sine and cosine functions and relationships. But any trigonometric function can be expressed in terms of the remaining functions, depending on the specific context. For instance, sine is the square root of the difference of one and cosine squared, while cosine is the square root of the difference of one and sine squared.
With algebraic formulae intact, cubic functions can be solved with Cardano’s formula for the determinants at a minimum, but the solutions are often difficult if not impossible to achieve. Quartic functions can also be solved using radicals; however, per the Abel-Ruffini theorem, quintic and higher-order functions almost never can be solved in such terms, with the simplest example being “x to the fifth minus x minus one equals zero.”
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