Approximate the Dottie number The Dottie number is the fixed point of the cosine function, or the solution to the equation cos(x)=x.1

Your task will be to make code that approximates this constant.  Your code should represent a function that takes an integer as input and outputs a real number.  The limit of your function as the input grows should be the Dottie number.

You may output as a fraction, a decimal, or an algebraic representation of a number.  Your output should be capable of being arbitrarily precise, floats and doubles are not sufficient for this challenge.  If your language is not capable of arbitrary precision numbers, then you must either implement them or choose a new language.

This is a code-golf question so answers will be scored in bytes, with fewer bytes being better.

Tips

One way of calculating the constant is to take any number and repeatedly apply the cosine to it.  As the number of applications tends towards infinity the result tends towards the fixed point of cosine.

Here is a fairly accurate approximation of the number.

0.739085133215161


1: Here we will take cosine in radians
MATL, 34 30 19 bytes
11 bytes off thanks to Sanchises!
48i:&quot;'cos('wh41hGY$

The last decimal figures in the output may be off. However, the number of correct figures starting from the left increases with the input, and the result converges to the actual constant.
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Explanation
For input n, and starting at x=1, this applies the function
              x ↦ cos(x)
with n-digit variable-precision arithmetic n times.
48         % Push 48, which is ASCII for '1': initial value for x as a string
i:&quot;        % Do n times, where n is the input
  'cos('   %   Push this string
  w        %   Swap. Moves current string x onto the top of the stack
  h        %   Concatenate
  41       %   Push 41, which is ASCII for ')'
  h        %   Concatenate. This gives the string 'cos(x)', where x is the
           %   current number
  GY$      %   Evaluate with variable-prevision arithmetic using n digits
           %   The result is a string, which represents the new x
           % End (implicit). Display (implicit). The stack contains the last x