Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

How to do it.

This problem appears in another thread:

We are asked to find the roots.

It is obvious that even M will have problems simplifying this so we can try a numerical attack. Mind you getting all the roots will be difficult since we do not have the luxury of even knowing how many there are.

The form given is already a recurrence so we will try that first.

We try:

x = .5;

x = ArcSin[2 Tan[x]^2] - (1/2) ArcTan[(3 Sin[2 x])/(4 Cos[2 x] + 5)]

Iterating this we find that x is approaching 0. We check and find that 0 is indeed a root.

We try again with another guess,

x = .7;

x = ArcSin[2 Tan[x]^2] - (1/2) ArcTan[(3 Sin[2 x])/(4 Cos[2 x] + 5)]

Iterating this appears to stabilize at

x = -1.570796326794897 - 1.945432927750411 i

We can immediately clean this up a bit with the ansatz that we are dealing with

we check and this too appears to be a root. Graphing shows that there are at least 2 real roots and to find another we use Newtons.

x = .5;

x = x - (x - ArcSin[2 Tan[x]^2] + (1/2) ArcTan[(3 Sin[2 x])/(4 Cos[2 x] + 5)])/(1 + ((6 Cos[2 x])/(5 + 4 Cos[2 x]) + (24 Sin[2 x]^2)/(5 + 4 Cos[2 x])^2)/(2 (1 + (9 Sin[2 x]^2)/(5 + 4 Cos[2 x])^2)) - (4 Sec[x]^2 Tan[x])/ Sqrt[1 - 4 Tan[x]^4])

Iterating this produces x = 0.5127879211591762, this is another real root.

So we have 3 roots at least, there are probably more...Can you find some?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: How to do it.

Where is the other thread? And this theead deserves a better title

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: How to do it.

I do not remember. The title says it all.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.