Step 1: Understanding the function $f(x) = \sin \left[ \cos^{-1} \left( x + \frac{2}{5} \right) \right]$

This function contains two operations:

• $\cos^{-1} \left( x + \frac{2}{5} \right)$, the inverse cosine (arc cosine), which gives an angle $\theta$ whose cosine is $x + \frac{2}{5}$.
• $\sin \left( \theta \right)$, the sine of that angle.

Step 2: Simplifying the expression

Let’s set: $\theta = \cos^{-1} \left( x + \frac{2}{5} \right)$ By the definition of inverse cosine, we know: $\cos(\theta) = x + \frac{2}{5}$ Now, using the Pythagorean identity for sine: $\sin(\theta) = \sqrt{1 - \cos^2(\theta)}$ Substitute $\cos(\theta) = x + \frac{2}{5}$: $\sin(\theta) = \sqrt{1 - \left( x + \frac{2}{5} \right)^2}$ Thus, the simplified form of $f(x)$ is: $f(x) = \sqrt{1 - \left( x + \frac{2}{5} \right)^2}$

Step 3: Finding the domain of $f(x)$

For the square root to be real, the argument inside the square root must be non-negative: $1 - \left( x + \frac{2}{5} \right)^2 \geq 0$ This simplifies to: $\left( x + \frac{2}{5} \right)^2 \leq 1$ Taking the square root of both sides: $-1 \leq x + \frac{2}{5} \leq 1$ Now, subtract $\frac{2}{5}$ from all parts of the inequality: $-1 - \frac{2}{5} \leq x \leq 1 - \frac{2}{5}$ Simplifying further: $-\frac{7}{5} \leq x \leq \frac{3}{5}$ Thus, the domain of $f(x)$ is: $\boxed{ \left[ -\frac{7}{5}, \frac{3}{5} \right] }$

Final Answer:

The simplified form of $f(x)$ is: $f(x) = \sqrt{1 - \left( x + \frac{2}{5} \right)^2}$ The domain of $f(x)$ is: $\boxed{ \left[ -\frac{7}{5}, \frac{3}{5} \right] }$

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the geometric interpretation of $\sin \left( \cos^{-1}(x) \right)$?
2. How does the inverse cosine function restrict the range of inputs?
3. Can you compute $f(0)$ for this function?
4. How would the domain change if $\frac{2}{5}$ were a different constant?
5. What are the properties of the arc-cosine function?

Tip: The inverse trigonometric functions, like $\cos^{-1}(x)$, always return angles in specific ranges, typically restricted by the properties of the corresponding trigonometric function.