Summary The Hyperbolic Functions in Calculus

CALCULUS

HYPERBOLIC FUNCTIONS
Hyperbolic functions are based on a hyperbola.




Hyperbolic Identities:
• Cosh(x)2 – sinh(x)2 = 1
• Sinh(-x) = -sinh(x) ~ ODD FUNCTION
• Cosh(-x) = cosh(x) ~ EVEN FUNCTION
• 1 – tanh(x)2 = sech(x)2
• 1 – coth(x)2 = -cosech(x)2
• Sinh(2x) = 2sinh(x)cosh(x)
• Cosh(2x) = cosh(x)2 + sinh(x)2
−1+𝑐𝑜𝑠ℎ(2𝑥)
• Sinh(x)2 =
2
1+𝑐𝑜𝑠ℎ(2𝑥)
• Cosh(x)2 =
2

Sum and Difference Formulas :

• Sinh (a + b) = sinh(a)cosh(b) + cosh(a)sinh(b)
• Sinh (a - b) = sinh(a)cosh(b) - cosh(a)sinh(b)
• Cosh (a + b) = cosh(a)cosh(b) + sinh(a)sinh(b)
• Cosh (a - b) = cosh(a)cosh(b) - sinh(a)sinh(b)

, Convert from Hyperbolic Function to Exponential Equation:
ⅇ 𝑥 −ⅇ −𝑥
❖ Sinh(x) =
2
ⅇ 𝑥 +ⅇ −𝑥
❖ Cosh(x) =
2
ⅇ 𝑥 −ⅇ −𝑥
❖ Tanh(x) =
ⅇ 𝑥 +ⅇ −𝑥

ⅇ 2𝑥 −1
=
ⅇ 2𝑥 +1

Convert from Exponential Equation to Hyperbolic Function:

➢ ⅇ 𝑥 = 𝑐𝑜𝑠ℎ(𝑥) + 𝑠𝑖𝑛ℎ(𝑥)
➢ ⅇ −𝑥 = cos ℎ(𝑥) − sin ℎ(𝑥)
1+tanh(𝑥)
➢ ⅇ 2𝑥 =
1−tanh(𝑥)
➢ 2𝑥
ⅇ = cosh(2𝑥) ⊢ sinh(2𝑥)
➢ (ⅇ 𝑥 )𝑛 = (cosh(𝑥) + sinh(𝑥))𝑛


ⅇ𝒙 −ⅇ−𝒙
Sinh(x) =
𝟐