When does Cos X = X. I want a "proof" please, I know it is at .739085133 radians but dont know "why".

Question

Here's what I know so far.

When measuring in radians, the angle measure is also the corresponding length of arc segment

(for Cos X = X)
Cos X = the length of the arc segment chopped by the angle theta

That means that the arc segment at .739 radians could be "unwound into a striaght line" and would be equal in length to the cos of .739 radians.

The cosine of .739 is the adjacent "leg" of the right triangle formed by the X axis, and the radius (as hypotenuse) from center of unit circle projected at .739 radians N of E until it hits the unit circle. Then dropping straight south.

Also I know that the sin of .739 radians is the absolute magniute of the upper endpoint of the arc segment from X axis.

I am very familiar with the visual understanding of trig and unit circle, including versine, exsecant, every trig function.

I still do not understand "why" the length of the unwound arc segment (into a striaght line) also equals the cosine for this radian angle measurement.

My best "approach" to my own figuring this question out, which albeit unsuccessful is this line of logic.

"If one considers ever possible arc starting from 1,0 of the unit circle in Quadrant 1"

Q1. when does the Y-value of the upper endpoint (the Y-value is obviously the sin for that angle which forms the arc) equal the legth of the chord segment

Q2. When is the length of the arc segment

(not the length of the chord)

equal to the X-value of the upper endpoint.

I feel that there is a rule which explains in its definition (such as area of a circle or length of a hypotenuse) which will make me "see" why Cos X = X at some point on the unit circle between 0 and Pi/2 (the first quadrant).

Answer 1

Sentriclelaw --

This is a very good question. I think what you're asking for is an analytical proof; such as a way to algebraically manipulate the symbols in the equation "Cos(x)=x" until you get "x = (something)," where "(something)" has no x's in it, and evaluates to .739.

Unfortunately, there is no such analytical proof. This is one of a large class of equations that can be expressed exactly, yet cannot be solved by manipulating the symbols. They can only be solved by so-called "numerical techniques," which basically means following a series of repetitive arithmetic steps to get increasingly accurate approximations.

Another way to put it is: The solution for x (in this case) cannot be expressed as a "formula" (a set of symbols); it can only be expressed as an "algorithm" (a sequence of steps).

One very good, and simple, numerical technique is called "Newton's method." If you haven't heard of it already, you'll learn about it when you study calculus. Newton's method works like this:

1. Start with an initial guess for the value of X. This is your "first approximation."
2. Plug the approximation into a special formula (the exact formula depends on the original equation you're trying to solve).
3. The result of the formula will be another approximation, but it will be "closer" than your previous approximation.
4. Take your new approximation, go to Step 2, and repeat.

You keep repeating Steps 2 through 4, getting better and better approximations. After a while (usually not too long), your approximation is so good that the value you get out of Step 4 is indistinguishable from the value you put into Step 2. At that point, you decide that your approximation is "close enough," and you stop.

To solve the particular equation "cos(x) = x", it turns out that the formula to use in Step 2 is the following:

A1 = A0 + (cos(A0) - A0) / (sin(A0) + 1)

where "A0" is the approximation that you plug into the formula; and "A1" is the (improved) approximation that you get out.

Doing Step 1 (making the initial guess) is not an exact science; and in some cases, if your initial guess is way off, Newton's method fails completely. That's all I'll say about that; you'll learn more in calculus.

So, let's try using Newton's method on your equation. It's pretty clear that the solution must lie somewhere between 0 and 1, so let's choose 0.5 as our initial guess. Here are the results we get (I wrote a short program to do this; you can duplicate it on your calculator):

Approximation 1: 0.5
Approximation 2: 0.7552224171056364
Approximation 3: 0.7391416661498792
Approximation 4: 0.7390851339208068
Approximation 5: 0.7390851332151607
Approximation 6: 0.7390851332151607

So you see, by the time we get to Approximation 6, the value is so close that it is no longer changing (at least to 16 digits of accuracy!) If we had chosen (for example) 0.1 or 0.9 as our initial guess, we might have had to go through the loop an extra time or two; but the approximations would still converge on 0.7390851332151607. Try it yourself!

Answer 2

> Why was my first solution not complete? because it didn't include pi/2 or 3*pi/2. > How can I make it complete? the way you did; by factoring instead of cancelling cos x. > Is there some rule against dividing by cos x (or sin x, tan x, etc)? there's the general rule that you can never divide by zero. if you ever divide by some function of x, you must ensure that the function cannot take on a value of zero before dividing. > Or do I need to consider if x is negative? aha. your question should say to find all solutions for x in the interval [0, 2*pi]. if not, your book's answer is wrong; there are infinitely many solutions, including infinitely many solutions of negative values of x. in particular, since sin(x + 2*pi) = sin x and cos(x + 2*pi) = cos x, if you have any particular solution x, then x + 2*pi is also a solution (and x - 2*pi). there may even be more (sorry, i haven't fully analyzed your problem). it's a good exercise to find *all* solutions, and write the solution set in a form mathematicians use. HTH & HAND!

Answer 3

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RE:
When does Cos X = X. I want a "proof" please, I know it is at .739085133 radians but dont know "why".
Here's what I know so far.

When measuring in radians, the angle measure is also the corresponding length of arc segment

(for Cos X = X)
Cos X = the length of the arc segment chopped by the angle theta

That means that the arc segment at .739 radians could be "unwound into a...

Answer 4

This is a nonlinear problem with no closed form solution, unless you want to define such a function that would be the inverse of the infinite series represented by cos(x)/x. You solve this numerically. If there was something to "prove", you would have to have a solution that wasn't just some numeric one.

Answer 5

Your use of quotes around "why" betrays your uncertainty as to what it is you seek. "how do I solve this equation for x analytically" is a clearer question. Not all equations can be solved analytically, and numerical methods, such as the one that gave you the answer, are required. Such is the case here.

Answer 6

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What is cos(pi/2), or cos(3/2 pi)? Hint: you can't divide by it.

Answer 7

X Cosx