“Instrumental Flying”

Consider the following set

S_1 = \{(x,y) | x=\frac{e^y-e^{-y}}{2}\}.\quad\quad\quad(1-1)

For all (x, y_1), (x, y_2) \in S_1, we have

\frac{e^{y_1}-e^{-y_1}}{2}=x, \frac{e^{y_2}-e^{-y_2}}{2}=x.

That is,

(x, y_1), (x, y_2) \in S_1 \implies \frac{e^{y_1}-e^{-y_1}}{2}- \frac{e^{y_2}-e^{-y_2}}{2}= 0.\quad\quad\quad(1-2)

Since \frac{e^{y_1}-e^{-y_1}}{2}- \frac{e^{y_2}-e^{-y_2}}{2}= 0 if and only if y_1 = y_2 (Exercise-1),

\frac{e^{y_1}-e^{-y_1}}{2}-\frac{e^{y_2}-e^{-y_2}}{2}=0 \implies y_1 = y_2.\quad\quad\quad(1-3)

From (1-2) and (1-3), we conclude:

\forall (x, y_1), (x, y_2) \in S_1 \implies y_1 = y_2.

i.e., S_1 defines a function.

Alternatively, (1-1) can be expressed as

S_1 = \{(x, y) | x = \sinh(y)\}.

It shows that S_1 is the inverse of \sinh(x). Therefore, we name the function defined by (1-1) \sinh^{-1} and write it as

y = \sinh^{-1}(x).


Let’s look at another set:

S_2 = \{(x,y)|x=\frac{e^y+e^{-y}}{2}\}.\quad\quad\quad(2-1)

For a pair (x, y_1>0) \in S_2, we have

x=\frac{e^{y_1}+e^{-y_1}}{2}.\quad\quad\quad(2-2)

For another pair (x, y_2=-y_1),

\frac{e^{y_2} + e^{-y_2}}{2} = \frac{e^{-y_1}+e^{-(-y_1)}}{2} = \frac{e^{-y_1} + e^{y_1}}{2} \overset{(2-2)}{=} x\implies (x, y_2=-y_1) \in S_2.

Since y_1>0, y_2 = -y_1 \neq y_1, (x, y_1), (x, y_2) \in S_1 does not implie y_1 = y_2. It means S_2 does not define a function.

However, modification of S_2 gives

S_3 = \{(x, y) | x=\frac{e^y+e^{-y}}{2}, y \ge 0\}.\quad\quad\quad(2-3)

It defines a function.

Rewrite (2-3) as

S_3 = \{(x, y) | x = \cosh(y), y \ge 0\}.\quad\quad\quad(2-4)

Then \forall (x, y_1), (x, y_2) \in S_3, we have

x=\cosh(y_1), x=\cosh(y_2)\implies \cosh(y_1) = \cosh(y_2).\quad\quad\quad(2-5)

Notice that by (2-3), y_1, y_2 \ge 0.

Cleary, (2-5) is true if and only if y_1 =y_2. For if y_1 \ne y_2, by (\star), \cosh(y_1) \ne \cosh(y_2).

Therefore,

(x, y_1), (x, y_2) \in S_3 \implies y_1 = y_2.

We name the function defined by (2-3) \cosh^{-1} as (2-4) shows that it is the inverse of \cosh(x).

See “Deriving Two Inverse Functions” for the explicit expressions of \sinh^{-1} and \cosh^{-1}.


Prove

t_1 \ge 0, t_2 \ge 0, t_1 \ne t_2 \implies \cosh(t_1) \ne \cosh(t_2).\quad\quad\quad(\star)

Without loss of generality, we assume that t_2 > t_1. By Lagrange’s mean-value theorem (see “A Sprint to FTC“),

\cosh(t_2) -\cos(t_1) =\cosh'(\xi) (t_2-t_1)

where \xi \in (t_1, t_2).

We have

t_2 > t_1 \implies t_2 - t_1 >0

and

\forall t > 0, (\cosh(t))' = (\frac{e^t+e^{-t}}{2})' = \frac{e^t-e^{-t}}{2}  \overset{t>0\implies t >-t \implies e^t  >e^{-t}}{>} 0.

Consequently,

\cosh(t_2) - \cosh(t_1) = \cosh'(\xi) (t_2-t_1) > 0.

i.e.,

\cosh(t_2) \ne \cosh(t_1).


Exercise-1 Show that \frac{e^{y_1}-e^{-y_1}}{2}- \frac{e^{y_2}-e^{-y_2}}{2}= 0 if and only if y_1 = y_2.

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