Hyperbolic Trigonometric Identity: sinh(x/2) + sign function sgn(x)

In this video I go over another hyperbolic trig identity and this time derive the half angle (or half argument) identity sinx(x/2) from my earlier double angle cosh(2x) identity. The derivation involves squaring rooting several terms and thus we need to account for the negative signs that might arise. I show that sinh changes values from negative to positive as x goes from less than to greater than 0. And since sinh(0) = 0, I show that it has the same characteristics as what is known as the sign or signum function, sgn(x). Thus we can write the identity for sinh(x/2) in just one expression instead of 3 by utilizing the sgn(x) function. This is a very interesting video in not only deriving hyperbolic trig but also in introducing the use of special functions such as sgn(x) to simplify mathematical expressions, so make sure to watch this video!

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Related Videos:

Hyperbolic Trigonometric Identity: cosh(2x) & sinh(2x): https://youtu.be/8d-CLGWFDH8
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Hyperbolic Trigonometric Identity: cosh(2x) Corollary 1 & 2 Formulas: https://youtu.be/A_juL30FU5k .

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