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Hyperbolic Trigonometry

In mathematics, we study trigonometry that deals with the right triangles and the measure of their sides and angles. The six trigonometric functions learnt in trigonometry are sin, cos, tan, csc, sec and cot.

There are various identities based on the trigonometric functions. These identities are widely used in solving trigonometric problems.

There is an extension of trigonometry which is known as hyperbolic trigonometry. It is said to be a variation in the trigonometry. It is concerned with the special kind of trigonometric functions, called as hyperbolic trigonometric functions.

The study about the problems based on hyperbolic functions is known as hyperbolic trigonometry. In this topic also, there are various hyperbolic identities utilized in hyperbolic trigonometric problems. In this page, we are going to learn about hyperbolic trigonometry and examples based on them.

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Hyperbolic Trigonometric Functions

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Hyperbolic trigonometric functions seem to be almost similar to ordinary trigonometric functions, but actually they are entirely different from ordinary trig functions. Hyperbolic trig function are sine hyperbolic, cosine hyperbolic, tan hyperbolic, csc hyperbolic, sec hyperbolic and cot hyperbolic.

These are denoted by sinh, cosh, tanh, cosech, sech and coth. These are defined in the context of rectangular hyperbola.

The sine hyperbolic and cosine hyperbolic functions are defined as shown in the following diagram :
Hyperbolic Trigonometry
All other four hyperbolic functions are based on these two. The relation between the six hyperbolic trig functions is exactly similar to the relationships between ordinary trigonometric functions; i.e. sinh and csch are reciprocal to each other,

$sinh$ = $\frac{1}{csch}$

cosh and sech are reciprocal to each other,

$cosh$ = $\frac{1}{sech}$

Also, tanh and coth are reciprocal to each other.

$tanh$ = $\frac{1}{coth}$

tanh and coth are defined in the form of ratios of sinh and cosh, i.e.

$tanh$ = $\frac{sinh}{cosh}$

$coth$ = $\frac{cosh}{sinh}$

Mathematically, the hyperbolic trigonometric functions are formulated in terms of exponential function, i.e. e$^{x}$.

1) $sinh\ x $= $\frac{e^{x}-e^{-x}}{2}$

2) $cosh\ x$ = $\frac{e^{x}+e^{-x}}{2}$

3) $tanh\ x$ = $\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$

4) $csch\ x$ = $\frac{2}{e^{x}-e^{-x}}$

5) $sech\ x$ = $\frac{2}{e^{x}+e^{-x}}$

6) $coth\ x$ = $\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$

Hyperbolic trigonometric Identities

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There are various different trigonometric identities defined in hyperbolic trigonometry. These are listed below :

Even and Odd Hyperbolic Functions:
sinh (-x) = - sinh x

ii) csch (-x) = - csch x

iii) cosh (-x) = cosh x

iv) sech (-x) = sech x

v) tanh (-x) = - tanh x

vi) coth (-x) = - coth x

Pythagorean Hyperbolic Identities:

i) $cosh^{2}x$ - $sinh^{2}x$ = 1

ii) $coth^{2}x$ - $csch^{2}x$ = 1

iii) $tanh^{2}x$ + $sech^{2}x$ = 1

Hyperbolic Addition Identities:

i) sinh (A $\pm$ B) = sinh A cosh B $\pm$ cosh A sinh B

ii) cosh (A $\pm$ B) = cosh A cosh B $\pm$ sinh A sinh B

Relationships Between Hyperbolic and Ordinary Trig Functions:

i) sinh x = - i sin(ix)

ii) cosh x = cos(ix)

iii) sech x = sec(ix)

iv) csch x = i csc(ix)

v) tanh x = -i tan(ix)

vi) coth x = i cot (ix)

Double Angle Hyperbolic Identities:

i) $sinh (2x) $= 2 $sinh x cosh x$

ii) $cosh (2x)$ = $cosh^{2}x$ + $sinh ^{2}x$

iii) $cosh (2x)$ = 2 $sinh^{2}x$ + 1

iv) $cosh (2x)$ = 2 $cosh^{2}x$ - 1

v) $tanh\ x$ = $\frac{2tanh\ x}{1+tanh^{2}x}$

Basic Differential Identities:

(i) $\frac{d}{dx}$ $sinh x$ = $cosh x$

(ii) $\frac{d}{dx}$ $cosh x$ = $sinh x$

(iii) $\frac{d}{dx}$ $tanh x$ = $sech^{2}x$

(iv) $\frac{d}{dx}$ $coth x$ = - $csch^{2}x$

(v) $\frac{d}{dx}$ $sech x$ = - $sech x tanh x$

(vi) $\frac{d}{dx}$ $csch x $= - $csch x coth x$

Basic Integral Identities:

i) $\int$ $sinh x dx$ = $cosh x$ + $c$

ii) $\int$ $cosh x dx$ = $sinh x$ + $c$

iii) $\int$ $csch^{2}x dx$ = - $coth x$ + $c$

iv) $\int$ $sech^{2}x dx$ = $tanh x$ + $c$

v) $\int$ $csch x coth x dx$ = - $csch x$ + $c$

vi) $\int$ $sech x tanh x dx$ = - $sech x$ + $c$

Solved Problems

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Some problems based on hyperbolic functions are illustrated below:

Problem 1: Find the simplified value of tanh(ln x).
Solution: tanh(ln x)

= $\frac{e^{ln\ x}-e^{-ln\ x}}{e^{ln\ x}+e^{-ln\ x}}$

= $\frac{e^{ln\ x}-e^{ln\( x^{-1})}}{e^{ln\ x}+e^{ln\ (x^{-1})}}$

= $\frac{e^{ln\ (x)}-e^{ln (\frac{1}{x})}}{e^{ln\ (x)}+e^{ln (\frac{1}{x})}}$

= $\frac{x- \frac{1}{x}}{x+ \frac{1}{x}}$

= $\frac{\frac{x^{2}-1}{x}}{\frac{x^{2}+1}{x}}$

= $\frac{x^{2}-1}{x^{2}+1}$

Problem 2: Prove that tanh (A + B) = $\frac{sinh A + cosh B}{1 + sinh A cosh B}$

tanh (A + B) = $\frac{sinh(A+B)}{cosh(A+B)}$

= $\frac{sinh A\ cosh B + cosh A\ sinh B}{cosh A\ cosh B + sinh A\ sinh B}$

On dividing by cosh A cosh B in numerator and denominator, we get

= $\frac{\frac{sinh A\ cosh B + cosh A\ sinh B}{cosh A\ cosh B}}{\frac{cosh A\ cosh B + sinh A\ sinh B}{cosh A\ cosh B}}$

= $\frac{tanh A + tanh B}{1 + tanh A\ tanh B}$
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