Hyperbolic functions of zero

Knowing the definitions that we covered of sinh, cosh and tanh, we’re able to work out some exact values of the hyperbolic functions - by substituting for x.

\sinh0=0

Proof:

• \sinh x=\frac{e^x-e^{-x}}2
• \sinh0=\frac{e^0-e^{-0}}2
• =\frac{1-1}2 (anything to the power of 0 is 1)
• =\frac02
• =0

\cosh0=1

Proof:

• \cosh x=\frac{e^x+e^{-x}}2
• \cosh0=\frac{e^0+e^{-0}}2
• =\frac{1+1}2 (anything to the power of 0 is 1)
• =\frac22
• =1

\tanh0=

Proof:

• \tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}
• \tanh 0=\frac{e^0-e^{-0}}{e^0+e^{-0}}
• =\frac{1-1}{1+1} (anything to the power of 0 is 1)
• =\frac02
• =0