An easy problem!

[DiGiTaIErRoR]

solve for x if:

arcsin(x)=cos(x)

accurate to 10 decimal places.

[Lior]

Hi,

When I tried to solve this equation I've noticed a few facts:

1. As X grows Cos(X) decreases and ArcSin(X) Increases so the equation actually claims that the two functions meet somewhere on the way.

2. Using the function ArcSin(x) means -1<X<1

3. Cos(-X) = Cos(+X) which means that both for positive and negative angle the cosine function always returns a positive value.

4. ArcSin(-X) = -ArcSin(X) which means that the returned value is positive only if the angle is positive and vice versa.

5. That's why there is no need to look for X<0, or in other words, the new more specefic range for X is now 0<X<1

6. If X>(Pi/180) Then ArcSin(X)>1 which exceeds the definition range of X.

7. Therefore, we can now be even more specific and say that X must be between 0<X<=(Pi/180)

That's what I concluded so far.

I guess we can now simply code a little program for scanning all numbers from 0 to Pi/180 for solving the equation ArcSin(X)=Cos(X) increasing the loop's variable each time by 10^(-10) or any other precision wanted.

As far as I know, VB does not support the ArcSin funtion, but anyway, you can calculate it through the supported ArcTan function, Called Atn(X) in VB:

ArcSin(X) = Atn(X / Sqr(1-X*X))

To convert degrees to radians multiply degrees by Pi/180.

To convert radians to degrees multiply radians by 180/Pi.


Hey..., On a second thought..., I guess I actually already told the answer without noticing it.

X=Pi/180, no?

Since Pi is irrational constant, it's a bit hard to post the answer decimally.

Have a good day.

[sql_lall]

Thanks for the formula (i was always wondering how to do that)

Anyway, just wondering if those equations were in degrees, radians, or Gradients (though i guess you *could* convert at the end i suppose)

Oh, and that increment won't work necessarily (almost definitely), as the number is more than likely to have square-roots in it somewhere. (though now i've said that, the answer will probably be 0.001 or something simple )

Anyway, that process (making the X bounds smaller by using the Arcsin part, then from this making the X bounds smaller by using the cos part) maybe can be used over and over again to find limits for X.

P.S. do you actually know the answer, or want to find it out??

[Lior]

Hi,

The equation I posted was in radians.

I dont know the answer, but I have some kind of intuition it's Pi/180.

We just need to calculate Pi far enough so it wont make any differences within the first 10 digits if is divided by 180.

[DiGiTaIErRoR]

The answer is not pi/180.

pi/180 = .01745329252

the arcsin of that being .017454178738

the cos of pi/180 is .999847695156

let me get you started:

cos(x)-arcsin(x)=0

and .5 < x < 1

[DiGiTaIErRoR]

No one has an answer?

[Lior]

Excuse me, but:

Arcsin(Pi/180) = 1.00005077653060952047020393467106

and not as you posted.

and Cos(Pi/180) = 0.99999995360411414000711804671704

I think this difference occurs only because pi is not calculated far enough.

But I will try to develop your way now too...

Talk to you later.

[Lior]

It's kind of weird, but using the way of simplifying expressions, It looks like X has no solution, as I will explain:

Since we use ArcSin(X) then -1<=X<=1

As we said above, Cos(-X) = Cos(X) and ArcSin(-X) = -ArcSin(X) so we can say now that 0<X<1 (X is obviously not 0 and not 1)

Now let's go to the actual work:

If ArcSin(X) = Cos(X) Then their derivatives are also equal:

ArcSin'(X) = Cos'(X)

1 / Sqrt( 1 - X²) = -Sin(X)

Sin(X) = -1 / Sqrt(1 - X²)

Now, remember we said that 0<X<1 ?
If so, Sin(X) is clearly positive, right?
If Sin(X) is positive, then the denominator Sqrt(1 - X²) must be negative (we got -1 at the numerator).
As you can see, the denominator is always positive because 0<X<1

In conclusion, we got one side of the equation being always positive, while the other side is always negative.

This definitely ruined my strong intuition X was Pi/180.

By the way, if you claim X does really equal to some number, Where was I mistaken above?

[Lior]

Well ???

[DiGiTaIErRoR]

I assure you X has a solution.

Here's a hint where:

.5 < x < .7

[DiGiTaIErRoR]

I was using radians. Yes in degrees you're right.

[DiGiTaIErRoR]

In degrees the answer would be within:

0 < x < .1

In degrees the answer is still not pi/180. Although pi/180 is close.

[Lior]

But tell me where I am mistaken, where I show X may not has any solution. (a few posts above)

[DiGiTaIErRoR]

Sin(X) = -1 / Sqrt(1 - X²)

Is not true.

Nor is:

1 / Sqrt( 1 - X²) = -Sin(X)

How you got these is beyond me.

[Lior]

It's known that the derivative of the ArcSin function is:

ArcSin'(X) = 1 / Sqrt( 1 - X²)

and the derivative of Cos(x) is:

Cos'(X) = -Sin(X)

when we compare them against each other, we come up with no solution for X.

Weird !!!

[RAEsquivelC]

You could try x = 0.69481 96907 29, or:
arcsin(x) = 0.76816 91567 34
cos(x) = 0.76816 91567 38

[cyborg]

tried to use my calculator to solve this and i got
x=0.017452405627727
my calc also showed that
pi/180=0.017453292519943
so its almost equal