Hyperbolic Functions Formulas

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1. Hyperbolic Functions

• sin x = \(\frac{e^{x}-e^{-x}}{2}\)
• cosh x = \(\frac{e^{x}+e^{-x}}{2}\)
• tanh x = \(\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\)
• coth x = \(\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\)
• cosech x = \(\frac{2}{e^{x}-e^{-x}}\)
• sech x = \(\frac{2}{e^{x}+e^{-x}}\)

2. Domain & Range of Hyperbolic Functions

Function	Domain	Range
sinh x	R	R
cosh x	R	[1, ∞ )
tanh x	R	(-1, 1)
coth x	R0	R – [-1, 1]
cosech x	R0	R0
sech x	R	(0, 1]

3. Formulae for Hyperbolic Function
(A) Square Formulae

• cosh² x – sinh² x = 1
• sech² x + tanh² x = 1
• coth² x – cosech² x = 1
• cosh² x + sinh² x = cosh 2x

(B) Expansion Formulae

• sinh (x ± y) = sinh x cosh y ± cosh x sinh y
• cosh (x ± y) = cosh x cosh y ± sinh x sinh y
• tanh (x ± y) = \(\frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}\)

(C) (i) sinh x + sinh y = 2 sinh \(\frac{x+y}{2}\) cosh \(\frac{x-y}{2}\)
(ii) sinh x – sinh y = 2 cosh \(\frac{x+y}{2}\) sinh \(\frac{x-y}{2}\)
(iii) cosh x + cosh y = 2 cosh \(\frac{x+y}{2}\) cosh \(\frac{x-y}{2}\)
(iv) cosh x – cosh y = 2 sinh \(\frac{x+y}{2}\) sinh \(\frac{x-y}{2}\)

(D) (i) sinh 2x = 2 sinh x cosh x
= \(\frac{2 \tanh x}{1-\tanh ^{2} x}\)

(ii) cosh 2x = cosh² x + sinh² x
= 2 cosh² x – 1
= 1 + 2 sinh² x
= \(\frac{1+\tanh ^{2} x}{1-\tanh ^{2} x}\)

(iii) tan 2x = \(\frac{2 \tanh x}{1+\tanh ^{2} x}\)

(E) (i) sinh 3x = 3 sinh x + 4 sinh³ x
(ii) cosh 3x = 4 cosh³ x – 3 cosh x
(iii) tan 3x = \(\frac{2 \tanh x+\tanh ^{3} x}{1+3 \tanh ^{2} x}\)

(F) (i) coshx + sinhx = e^x
(ii) cosh x – sinh x = e^-x
(iii) (cosh x + sinh x)ⁿ = cosh nx + sinh nx

4. Relation between Hyperbolic and Circular Function
(i) sin (ix) = i sinh x
sinh (ix) = i sin x
sinhx = – i sin (ix)
sin x = – i sinh (ix)

(ii) cos (ix) = cosh x
cosh (ix) = cos x
cosh x = cos (ix)
cos x = cosh (ix)

(iii) tan (ix) = i tanh x
tanh (ix) = i tan x
tanh x = – i tan (ix)
tan x = – i tanh (ix)

(iv) cot (ix) = – i coth x
coth (ix) = – i cot x
coth x = i cot (ix)
cot x = i coth (ix)

(v) sec (ix) = sech x
sech (ix) = sec x
sech x = sec (ix)
sec x = sech (ix)

(vi) cosec (ix) = – i cosech x
cosech (ix) = – i cosec x
cosech x = i cosec (ix)
cosec x = i cosech (ix)

5. Period of Hyperbolic Functions
Period of sinh x = 2πi ; cosh x = 2πi and tanh x = πi

6. Relation between Inverse Hyperbolic Function and Inverse Circular Function

• sinh^-1 x = – i sin^-1 (ix)
• cosh^-1 x = – i cos^-1 x
• tan^-1 x = – i tan^-1 (ix)
• coth^-1 x = i cot^-1 (ix)
• sech^-1 x = – i sec^-1 x
• cosech^-1 x = i cosec^-1 (ix)

7. Relation between Inverse Hyperbolic Functions and Logarithmic Functions

• sinh^-1 x = log(x + \(\sqrt{x^{2}+1}\))(-∞ < x < ∞)
• cosh^-1 x = log (x + \(\sqrt{x^{2}-1}\)) (x ≥ 1)
• tanh^-1 x = \(\frac{1}{2}\) log \(\left(\frac{1+x}{1-x}\right)\)(|x| < 1)

Note: Formulae for values of cosech^-1 x, sech^-1 x and coth^-1 x may be obtained by replacing x by 1/x in the values of sinh^-1 x , cosh^-1 x and tanh^-1 x respectively.

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