Points of Intersection of Cosh(x) and Sec(x)

Question

How can I find the points of intersection of cosh(x) and sec(x)?

Plotting the two, I can see that it has a root around 4.7, but then I can't tell where the next root is.

But when I use FindRoot with a guess of 4.7 I don't get the root but if I use a guess between 4.72 and 4.8 then I find the root - 4.730040744862704.

Also, another root - 7.8532046241 can be found by guessing between 7.83 and 7.85.

Q: Is there a better way to solve the equation: Cosh[x]==Sec[x]? Q: How do I see the other points of intersection of the graphs y=Cosh[x] and y=Sec[x]?

Any help would be much appreciated. Thanks!

Answer 1

You can use Solve or Reduce if you add a domain restriction:

N @ Solve[Cosh[x] == Sec[x] && -10<x<10, x]
N @ Reduce[Cosh[x] == Sec[x] && -10<x<10, x]

{{x -> 0.}, {x -> 0.}, {x -> 0.}, {x -> 0.}, {x -> -7.8532}, {x -> -4.73004}, {x -> 4.73004}, {x -> 7.8532}}

x == -7.8532 || x == -4.73004 || x == 0. || x == 4.73004 || x == 7.8532

(using NSolve directly seems to miss roots)

Answer 2

From: About multi-root search in Mathematica for transcendental equations

Clear[findRoots]
Options[findRoots] = Options[Reduce];
findRoots[gl_Equal, {x_, von_, bis_}, 
prec : (_Integer?Positive | MachinePrecision | Infinity) : 
MachinePrecision, wrap_: Identity, opts : OptionsPattern[]] := 
Module[{work, glp, vonp, 
bisp}, {glp, vonp, bisp} = {gl, von, bis} /. 
r_Real :> SetPrecision[r, prec];
work = wrap@Reduce[{glp, vonp <= x <= bisp}, opts];
work = {ToRules[work]};
If[prec === Infinity, work, N[work, prec]]];

A domain restriction is -10 < x < 10

findRoots[Cosh[x] == Sec[x], {x, -10, 10}]

(* {{x -> -7.8532}, {x -> -4.73004}, {x -> 0.}, {x -> 4.73004}, {x -> 7.8532}} *)

Or:

findRoots[Cosh[x]*Cos[x] == 1, {x, -10, 10}](* 33 times faster *)

(* {{x -> -7.8532}, {x -> -4.73004}, {x -> 0.}, {x -> 4.73004}, {x -> 7.8532}}*)