# What happens if I use #cos# repeatedly?

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like if I use cos(cos(cos(cos(cos(cos(x)))))) upto a hundred times?

like if I use cos(cos(cos(cos(cos(cos(x)))))) upto a hundred times?

##### 1 Answer

It converges to the Dottie number.

#### Explanation:

The value will converge to the Dottie number

If we graph

graph{(y-x)(y-cos(x))=0 [-10, 10, -5, 5]}

There is no known closed form for the Dottie number, but, as asked, it can be approximated by repeatedly applying the cosine function to any given value (although there are other sequences which converge to it more quickly). In other words, it is the limit as

Another way of looking at it is to note that

If we did this many times, we would note that the rays would start to converge. The