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4 Elementary FunctionsHyperbolic Functions

§4.30 Elementary Properties

Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.
sinhθ=a coshθ=a tanhθ=a cschθ=a sechθ=a cothθ=a
sinhθ a (a2-1)1/2 a(1-a2)-1/2 a-1 a-1(1-a2)1/2 (a2-1)-1/2
coshθ (1+a2)1/2 a (1-a2)-1/2 a-1(1+a2)1/2 a-1 a(a2-1)-1/2
tanhθ a(1+a2)-1/2 a-1(a2-1)1/2 a (1+a2)-1/2 (1-a2)1/2 a-1
cschθ a-1 (a2-1)-1/2 a-1(1-a2)1/2 a a(1-a2)-1/2 (a2-1)1/2
sechθ (1+a2)-1/2 a-1 (1-a2)1/2 a(1+a2)-1/2 a a-1(a2-1)1/2
cothθ a-1(a2+1)1/2 a(a2-1)-1/2 a-1 (1+a2)1/2 (1-a2)-1/2 a