Question

Find the range of the function $f\left( x \right)=\sin \left( \cos x \right)$

Answer 1

Hint: You know that the trigonometric sine and cosine functions are periodic functions. The range of the trigonometric sine and cosine function is $\left[ -1,1 \right]$, where -1 is the minimum value and 1 is the maximum value.

Complete step-by-step answer:
Let the given function be defined as $f:R\to A$ such that$f(x)=\sin (\cos x)$.

We know that the range of $\sin x$ and $\cos x$ is $\left[ -1,1 \right]$ .

Hence, the range of $\sin \left( \cos x \right)$ will be located in the interval [−1, 1]. However, the required range of the given trigonometric function is denoted A.

\[A=\left[ \sin ({{x}_{1}}),\sin ({{x}_{2}}) \right]\]

Where $\sin ({{x}_{1}})$is the minimal sine value in the interval [−1, 1] and \[\sin ({{x}_{2}})\]

is the maximum value. As $\sin \left( x \right)$ is periodic function, so will$\sin \left( \cos x

\right)$.

Therefore \[A=\left[ \sin (-1),\sin (1) \right]\]

We know that, the negative angle of the sine function $\sin (-\theta )=-\sin \theta $

Hence \[A=\left[ -\sin (1),\sin (1) \right]\]

Therefore, the range of the given trigonometric function is\[\left[ -\sin (1),\sin (1) \right]\].


Note: You might get confused about the difference between the domain of a function and the range of a function. Domain is the independent variable and range is the dependent variable. On the other hand, range is defined as a set of all probable output values. Domain is what is put into a function, whereas range is what is the result of the function with the domain value