# Range of a function f(x)=cos(k sin x)is [-1,1] then the least positive integral value of k will be?

May 30, 2018

$k \ge 4$

#### Explanation:

The range of $k \sin x$ is $\left[- k , k\right]$.

As for $x = 0$,

$\cos \left(k \sin \left(x\right)\right) {|}_{x = 0} = 1$

The range includes $y = 1$ for any value of $k$.
On the other hand:

$\cos \left(k \sin x\right) = - 1 \iff k \sin x = \pi + 2 n \pi$

For $n = 0$:

$k \sin x = \pi$

and as

$k \sin x \le k$

Then $k$ must be greater that $\pi$, which means $k \ge 4$.