User:Jake Kump/sandbox

George Osborn (1864-1932) was an English mathematician, known for Osborn’s rule that deals with Hyperbolic Trigonometric identities.

Life
George Osborn was born in 1864 in Manchester, England. He attended Emmanuel College, Cambridge University in 1884. Where in 1887 he received the 17th Wrangler award for achieving a 1st in his mathematics degree. He then attended The Leys school, Cambridge in 1888. In 1891 George Osborn became assistant headmaster and senior science master at The Leys School. He continued to work at the school until his retirement in 1926. Apart from mathematics, George Osborn took his time to study the New Testament owing to his grand farther Revenant George Osborn the president of the Methodist Conference in 1863 and 1881. In addition to this, George Osborn enjoyed reading Spanish literature and was an avid chess player up until his death on October 14th, 1932.

Work
From 1902-1925 George Osborn would write numerous articles for The Mathematical Gazette. In his submissions he covered a range of topics from sums of cubes to series expansions. However, his most notable entry was in July of 1902 titled: Mnemonic for hyperbolic formulae. In this publication George Osborn outlined a rule, that he found useful for teaching, when converting between trigonometric and hyperbolic trigonometric identities. In conjunction with this George Osborn published various books with his colleague Charles Henry French, who was head of mathematics at The Leys School Cambridge. The titles of their joint work include: Elementary Algebra, First Year’s Algebra and The Graphical Representation of Algebraic Functions.

Osborn's Rule
Osborn’s Rule which was outlined by George Osborn in his 1902 Mathematical Gazette publication: Mnemonic for hyperbolic formulae. Aids in the conversion between trigonometric and hyperbolic trigonometric identities. To convert a trigonometric identity to the equivalent hyperbolic trigonometric identity, Osborn’s rule states to first write out all the cosine and sine compound and multiple angles terms to their constituent expanded parts. Then exchange all the cosine and sine terms to cosh and sinh terms. However, for all products or implied products of two sine terms replace it with the negative product of two sinh terms. This is because $$i\sin(ix)$$ is equivalent to $$\sinh(x)$$ and $$i^2$$ is equivalent to $$-1$$, so when multiplied to together the sign switched when compared to the regular trigonometric identity. Due to this fact however, for terms which have a product of a multiple of four sinh terms the sign does not change when compared to the regular trigonometric identity.

Trigonometric Identity:

$$\cos^2(x)+\sin^2(x)=1$$

$$1+\tan^2(x)=\sec^2(x)$$

$$\cot^2(x)+1=\csc^2(x)$$

$$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$

$$\cos(2x)=1-2\sin(x)$$

Hyperbolic Trigonometric Identity:

$$\cosh^2(x)-\sinh^2(x)=1$$

$$1-\tanh^2(x)=\operatorname{sech^2}(x)$$

$$-\coth^2(x)+1=-\operatorname{csch^2}(x)$$

$$\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)$$

$$\cosh(2x)=1+2\sinh(x)$$