# Do "other" trigonometric functions than Tan Sin Cos and their derivatives exist?

## 4 Answers By Expert Tutors

There are hyperbolic functions sinh, cosh, tanh, sech, csch and coth.

Mark M. answered • 03/21/19

Mathematics Teacher - NCLB Highly Qualified

They are the hyperbolic functions:

sinh = (e^{x} - e^{-x}) / 2

cosh = (e^{x} + e^{-x}) / 2

They have derivatives, tanh, csch, coth, and sech analogous to the other trig functions.

Yes, other Trig functions and their derivatives exist.

The first six are:

d/dx sin(x) = cos(x)

d/dx cos(x) = - sin(x)

d/dx tan(x) = sec^{2}(x)

d/dx sec(x) = sec(x)tan(x)

d/dx csc(x) = - csc(x)cot(x)

d/dx cot(x) = - csc^{2}(x)

I like to group them as follows, with those without "co" in the name in the 1st column and those with "co" in the 2nd. We can see patterns to help memorize them better this way.

d/dx sin(x) = cos(x) d/dx cos(x) = - sin(x)

d/dx sec(x) = sec(x)tan(x) d/dx csc(x) = - csc(x)cot(x)

d/dx tan(x) = sec^{2}(x) d/dx cot(x) = - csc^{2}(x)

(This is intended to be in two columns, but the Wyzant software keeps taking the space out)

The trig functions are based off of a unit circle, with equation x^{2} + y^{2} = 1

Additionally there are 6 hyperbolic functions, which are based off of the hyperbola x^{2} - y^{2} = 1

Their derivatives are:

d/dx sinh(x) = cosh(x)

d/dx cos(x) = sinh(x)

d/dx tan(x) = sech^{2}(x)

d/dx sech(x) = - sech(x)tanh(x)

d/dx csch(x) = - csch(x)coth(x)

d/dx coth(x) = - csch^{2}(x)

Finally, all 12 of the functions above have inverse functions, and each of those have derivatives as well. They are much more complicated to write out, as most have a numerator, denominator, and root sign, so I would suggest a quick web search for "inverse trig derivatives" and inverse hyperbolic derivatives".

Hope that helps.

You might be thinking of hyperbolic sine (sinh), hyperbolic cosine (cosh), etc which, although treated similar to trig functions, are actually exponential (e^{x}) functions.

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Kevin S.

03/14/19