Hyperbolic sine. Related to cosh; in fact, its function looks very similar:

(ex-e-x)/2

It is pronounced like "cinch" (think 'sin h', where sin is said like the religious term for wrongdoing, and h is said as a letter). A joke I thought was funny was to say "it's a sinh" and spell "it's a" with "iota, tau, sigma, alpha." The hyperbolicness of its context may be discovered in calculus. Not quite as useful a function as sine, but still helpful in certain cases. sinh is the derivative of cosh, and vice versa, unlike their cousins, cos and sin, where cos is the derivative of sin, but -sin is the derivative of cos.

In the UK, this function is often pronounced 'shine'. The reason for this is unclear: after all, we don't pronounce cosh as 'chose'.

I have always pronounced it 'shine'. I really had no formal exposure to it, instead learning about hyperbolic functions on my own. cosh=coshine made sense to me, and thus sinh=cosh-co=coshine-co=shine. 'cinch' has seemed contrived because of the extra 'c' sound.

By the way, I normally pronounce tanh 'thangent' for the same reason. On the other hand, if I had to pronounce sech, I'd say 'sechant' because 'ch' makes sense as a sound, unlike 'nh'.