**HP Prime: Hyperbolic CAS Transformations**

**Introduction**

These CAS transforms some expressions involving hyperbolic functions, mainly sinh (hyperbolic sine) and cosh (hyperbolic cosine).

Let ϕ and Ω be any algebraic expression, real number, or complex number. These commands are meant to work in CAS mode.

**Exponential Definitions**

**sinhexp**

sinhexp(ϕ) = (e^(ϕ) - e^(-ϕ)) / 2 = ((e^ϕ)^2 - 1) / (2 * e^ϕ)

#cas

sinhexp(f):=

BEGIN

RETURN (e^(f)-e^(−f))/2

END;

#end

**coshexp**

coshexp(ϕ) = (e^(ϕ) + e^(-ϕ)) / 2 = ((e^ϕ)^2 + 1) / (2 * e^ϕ)

#cas

coshexp(f):=

BEGIN

RETURN (e^(f)+e^(−f))/2

END;

#end

**tanhexp**

tanhexp(ϕ) = (e^(ϕ) - e^(-ϕ)) / (e^(ϕ) + e^(-ϕ))

#cas

tanhexp(f):=

BEGIN

RETURN (e^(f)-e^(−f))/

(e^(f)+e^(−f))

END;

#end

**Adding Properties**

**addsinh**

addsinh(ϕ + Ω) = sinh ϕ * cosh Ω + sinh Ω * cosh ϕ

#cas

addcosh(f,g):=

BEGIN

RETURN COSH(f)*COSH(g)+

SINH(f)*SINH(g);

END;

#end

**addcosh**

addcosh(ϕ + Ω) = csoh ϕ * cosh Ω + sinh Ω * sinh ϕ

#cas

addsinh(f,g):=

BEGIN

RETURN SINH(f)*COSH(g)+

COSH(f)*SINH(g);

END;

#end

**addtanh**

addtanh(ϕ + Ω) = (tanh ϕ + tanh Ω) / (1 + tanh ϕ * tanh Ω)

#cas

addtanh(f,g):=

BEGIN

RETURN (TANH(f)+TANH(g))/

(1+TANH(f)*TANH(g));

END;

#end

**Squaring Properties**

**sqsinh**

sqsinh(ϕ) = sinh^2 ϕ = 1/2 * cosh(2 * ϕ) - 1/2

#cas

sqsinh(f):=

BEGIN

RETURN COSH(2*f)/2-1/2;

END;

#end

**sqcosh**

sqcosh(ϕ) = cosh^2 ϕ = 1/2 * cosh(2 * ϕ) + 1/2

#cas

sqcosh(f):=

BEGIN

RETURN COSH(2*f)/2+1/2;

END;

#end

**Product Properties**

**sinhsinh**

sinhsinh(ϕ, Ω) = 1/2 * (cosh(ϕ + Ω) - cosh(ϕ - Ω))

#cas

sinhsinh(f,g):=

BEGIN

RETURN 1/2*(COSH(f+g)-

COSH(f-g));

END;

#end

**coshcosh**

coshcosh(ϕ, Ω) = 1/2 * (cosh(ϕ + Ω) + cosh(ϕ - Ω))

#cas

coshcosh(f,g):=

BEGIN

RETURN 1/2*(COSH(f+g)+

COSH(f-g));

END;

#end

**sinhcosh**

sinhcosh(ϕ, Ω) = 1/2 * (sinh(ϕ + Ω) + sinh(ϕ - Ω))

#cas

sinhcosh(f,g):=

BEGIN

RETURN 1/2*(SINH(f+g)+

SINH(f-g));

END;

#end

Source:

Spiegel, Murray R. and Seymour Lipschutz, John Liu.

__Schuam's Outlines: Mathematical Handbook of Formulas and Tables__5th Edition McGraw Hill: New York 2018 ISBN 978-1-260-01053-4

A little early start to our Thanksgiving feast,

Eddie

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