Hype for the Future 68J: The Derivatives of the Different Trigonometric Functions?

Overview

While novaTopFlex would like to explain the derivatives and integrals of the different trigonometric functions, LaTeX would be required in order to correctly interpret the underlying situation.

Hype for the Future 9B: Verfunctions in Trigonometry

The graph attached below showcases eight distinct and antiquated trigonometric functions regarding “versed” functions. Haversine in particular has historically been of importance in navigation, sailing, and perhaps even in communities such as fishing and whaling as well.

From top to bottom, the functions listed in the legend correspond with the versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, and hacovercosine, respectively.

Why is arctan(1/2) equal to arctan(1/3) + arctan(1/7)?

The truth behind the signature arctangent question is simply that the arctangent addition formula, in direct resemblance to the tangent addition formula, is directly related to the fact that 1/3 + 1/7 equals 7/21 + 3/21 and the fact that 7 + 3 = 10, resulting in a final solution of 10/21 just in the numerator. In the denominator, however, is 1 - (1/3)(1/7), which results in 1 - 1/21, which in return results in (21-1)/21 equals 20/21. If the numerator is 10/21 and the denominator is 20/21, the ratio is exactly identical to the ratio with numerator 10 and denominator 20. Furthermore, 10/20 is equivalent to the simplified expression 1/2.

Related Question: Why can Pi be represented by arctan(1)+arctan(2)+arctan(3)?

The arctangent of one plus the arctangent of two is equivalent to the arctangent of (1+2)/(1-1*2), which is the arctangent of 3/-1 equals -3. With two and three, the formula becomes (2+3)/(1-2*3), which becomes 5/-5 and thus becomes -1. For one and three, the formula is (1+3)/(1-1*3), which equates to 4/-2 equals -2. Multiplying the arctangents of the negatives by the multiplicative inverses effectively cancels out the numerators, as in x+(-x) equals x-x equals 0 (zero). Even though the arctangent of zero is, in fact, zero, the tangent function is notable for having a period of π, and with the positive arctangents, arctan(1)+arctan(2)+arctan(3) can, in fact, equal π.

Interesting Mathematics Facts

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.”