hyperbolic trig functions

further pure two, chapter four, hyperbolic functions

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Even and odd functions

Even function:

"for all real numbers x in its domain, f(x) = f(-x)"

Odd function:

"for all real numbers x in its domain, f(x) = -f(-x)"

Symmetrical domain:

"for these definitions to be valid, the demain D of f has to be symmetrical about 0: if x is in the domain, so is -x"

Theorem:

"any function f with a symmetrical domain can be written as the sum of an even function g and and odd funtion h"

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hyperbolic functions

The most important application of the theorem is when f is taken to be e^x

coshx = 1/2(e^x + e^-x)

sinhx = 1/2(e^x - e^-x)

When x is a large positive number, coshx ~ sinhx ~ 1/2e^x

When x is negative and modx is large, coshx ~ sinhx ~ 1/2e^-x

The range of sinh is R, however the range of cosh is y E R, y >(or equal to) 1

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hyperbolic and trigonometric functions

The basic identity:

cosh^2 x - sinh^2 x = 1

Addition Formulae:

cosh(A + B) = coshAcoshB + sinhAsinhB

sinh(A + B) = sinhAcoshB +coshAsinhB

cosh(A - B) = coshAcoshB - sinhAsinhB

sinh(A - B) = sinhAcoshB - coshAsinhB

cosh2A = cosh^2 A +sinh^2 A = 1 + 2sinh^2 A = 2cosh^2 A - 1

sinh2A = 2sinhAcoshA

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Differentiation and Integration

Differentiation:

d/dx coshx = sinhx

d/dx sinhx = coshx

Integration:

S sinhx dx = coshx + k

S coshx dx = sinhx + k

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Other Hyperbolic Functions

sechx = 1/coshx tanhx = sinhx/coshx

cosechx = 1/sinhx cothx = 1/tanhx = coshx/sinhx

Identity:

1 - tanh^2 x = sech^2 x

Differentiation:

d/dx tanhx = sech^2 x

d/dx sechx = -sechxtanhx

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Inverse Hyperbolic Functions

For all x:

sinh^-1 x = ln(x + root(1 + x^2))

For x > (or equal to) 1:

cosh^-1 x = ln(x + root(x^2 - 1))

For -1 < x < 1:

tanh^-1 x = 1/2 ln (1 +x/1 - x)

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Differentiation and Integration of Inverse Hyperbo

Differentiation:

d/dx cosh^-1 x = 1/root(x^2 - 1)

d/dx sinh^-1 x = 1/root(1 + x^2)

d/dx tanh^-1 x = 1/1 - x^2

Integration:

S 1/root(x^2 - a^2) dx = cosh^-1 x/a + k

S 1/root(a^2 + x^2) dx = sinh^-1 x/a + k

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