Hyperbolic functions

sinh(x)=

• $\frac{1}{2\ }\left(e^x+e^{-x}\right)$

• $\frac{1}{2\ }\left(e^x-e^{-x}\right)$

What is the exact value of cosh(ln2)

• $\frac{3}{4}$

• $\frac{5}{4}$

• $\frac{3}{2}$

• $5$

coshx+sinhx=

• $e^x+e^{-x}$

• $e^{-x}$

• $2e^x$

• $e^x$

 $\cosh^2A=$  

•  $1+\sinh^2A$  

•  $1-\sinh^2A$  

•  $\sinh^2A-1$  

Osborn’s rule is a rule for converting a trigonometric identity into a corresponding hyperbolic one. The rule states that one replaces every occurrence of sine or cosine with the corresponding hyperbolic sine or cosine, and wherever one has a product of two sines, the product of the hyperbolic sines must be negated.
For example,  becomes  $\cos2A=2\cos^2A-1$  becomes  $\cosh2A=2\cosh^2A-1$  while  $\cos2A=1-\sin^2A$  becomes  $\cosh2A=1+2\sinh^2A$  .

Given that  $\csc^2A=1+\cot^2A$  , from Osborn's rule find  $\operatorname{csch}^2A$  .

•  $1+\coth^2A$  

•  $1-\coth^2A$  

•  $\coth^2A-1$  

Solve the equation $4\cosh x+e^x=7$  

• x=ln2, x=ln3

• x=ln2, x=-ln3

• x=-ln2, x=ln3

• x=-ln2, x=-ln3

Given $\sinh x=\frac{4}{3}$  then  $\text{coshx}=$  

• 5/3

•  $\pm\frac{5}{3}$  

•  $\frac{\sqrt{7}}{3}$  

• No solution

The solutions to $1+\cosh^2x=3\sinh x$  are given by

•  $\sinh x=1\ \text{or }2$  

•  $\sinh x=-1\ \text{or }2$  

•  $\sinh x=1\ \text{or }-2$  

•  $\sinh x=-1\ \text{or -}2$  

 $\coth^{-1}(x)=$  

•  $\frac{1}{\tanh^{-1}\left(x\right)}$  

•  $\tanh^{-1}\left(\frac{1}{x}\right)$  

 $\operatorname{sech}^{-1}\left(x\right)=$  

•  $\cosh^{-1}\left(\frac{1}{x}\right)$  

•  $\frac{1}{\cosh^{-1}\left(x\right)}$