So one can clearly see that there is exactly one solution. We can use Bolzano's theorem to actually prove that there must be at least one solution: Let h(x) = cos(x) - x. Then h is a continuous function and

• h(- ) = -1 + > 0
• h( ) = -1 - < 0

Hence, by Bolzano's theorem there must be at least one place where h( ) = 0, or equivalently where cos( ) = .

One can use Bolzano's theorem to construct an algorithm that will find zeros of a function to a prescribed degree of accuracy in many cases. In simple terms:

• start with an interval [a , b] where h(a) * h(b) < 0 (i.e. h(a) and h(b) have opposite signs)
• find a point c - usually (a + b) / 2 - such that either h(a) * h(c) < 0 or h(b) * h(c) < 0
  • if h(a) * h(c) < 0, repeat this procedure with b replaced by c
  • if h(b) * h(c) < 0, repeat this procedure with a replaced by c.
• Continue until the difference b - a is small enough.

Would this procedure find the zero of the function f(x) = in the interval [-1, 1] ?