Question

Find the range of f (x) = sin(cosx).​

Answer 1

Step-by-step explanation:

Let f:R→A, f(x)=cos(sinx).

There are multiple ways to go about this. We can use the fact that f is a composed function or do it directly. We'll use the second approach.

We know that the range of sinx and cosx is [−1,1]. Hence, The range of cos(sinx) will be located in the interval [−1,1]. However, the exact range, denoted A in this answer, is going to be:

A=[cos(x1),cos(x2)]

Where cos(x1) is the minimal cosine value in the interval [−1,1] and cos(x2) is the maximum value. As cos(x) is periodic, so will cos(sinx).

On the interval [−1,0] the cosine function is monotonic increasing, hence the minimal value of cosx is going to be when x=−1. Similarly, as the cosine is monotic decreasing on [0,1], this means x=0 is the max value and it also means that x=1 is also the minimal value.

∴A=[cos(−1),cos0]=[cos1,cos0]=[cos1,1]

Thus, the range is

A=[cos 1,1]