What is the value of a?help please?

Question

What is the value of a?help please?

if a is a positive number so that cosh a= 5/4 using only the definitions of the hyperbolic functions, compute the values of sinh a, tanh a, and sech a???


Additional Details:
cosh is a hyperbolic function, it is not cos

Answer 1

If cosh (a) = 5/4, then a = +/- ln 2 = +/- 0.693147181... ,

and sinh (a) = +/- 3/4, tanh (a) = +/- 3/5, and sech (a) = 4/5, where the signs of ' a,' sinh (a) and tanh (a) are the same.

Here's how to derive these results:

cosh a = cos (ia) = [e^a + e^(-a)] / 2. So, if cosh a = 5/4,

[e^a +e^(-a)] / 2 = 5/4, so then e^(2a) - (5/2) e^a + 1 = 0, therefore:

e^a = [5/2 +/- sqrt (25/4 - 4)]/ 2 = [5/2 +/- sqrt (9/4)]/ 2 = [5/2 +/- 3/2]/ 2 = 2 or 1/2.

So a = ln 2 or - ln 2 = +0.693147181... or -0.693147181... .

[These are +/- 0.693147181... as cosh x = cosh (-x) for all x, cosh (x) being an EVEN function of x.]

Now sinh (ln 2) = [e^(ln 2) - e^(-ln 2)] / 2 = [2 - 1/2]/ 2 = + 3/4,
but also sin (- ln 2) = - 3/4, sinh (x) being ODD for all x.

Then tanh (ln 2) = sinh (ln 2) /cosh (ln 2) = + (3/4) / (5/4) = + 3/5,
or tanh (- ln 2) = sinh (- ln 2) / cosh (- ln 2) = - (3/4) / (5/4) = - 3/5,

and sech (a) = 1 / cosh (a) = 4/5.

So if cosh (a) = 5/4, then a = +/- ln 2 = +/- 0.693147181... ,
sinh (a) = +/- 3/4, tanh (a) = +/- 3/5, and sech (a) = 4/5, where the signs of ' a,' sinh (a) and tanh (a) are the same.

QED

Live long and prosper.

Answer 2

http://www.sosmath.com/trig/hyper/hyper01/hyper01.html
cosh^2 -sinc^2 = 1