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To simplify the expression \( f(x) = \sin \left( \cos^{-1} \left( \frac{x + 2}{5} \right) \right) \) using a right triangle, we can proceed by considering the properties of trigonometric functions and their inverses.

### Step 1: Understanding the Inverse Cosine

Let:
\[
\theta = \cos^{-1} \left( \frac{x + 2}{5} \right).
\]

This means:
\[
\cos(\theta) = \frac{x + 2}{5}.
\]

### Step 2: Construct a Right Triangle

We can interpret this cosine value in a right triangle:
- The adjacent side to the angle \(\theta\) is \(x + 2\).
- The hypotenuse of the triangle is \(5\).

Now, using the Pythagorean theorem, the opposite side \( \text{opposite} \) can be calculated as:

\[
\text{opposite} = \sqrt{\text{hypotenuse}^2 - \text{adjacent}^2} = \sqrt{5^2 - (x + 2)^2} = \sqrt{25 - (x + 2)^2}.
\]

### Step 3: Finding \( \sin(\theta) \)

Now, using the triangle, we can find \( \sin(\theta) \):

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{25 - (x + 2)^2}}{5}.
\]

### Step 4: Simplifying \( f(x) \)

Since \( \theta = \cos^{-1} \left( \frac{x + 2}{5} \right) \), we have:

\[
f(x) = \sin \left( \cos^{-1} \left( \frac{x + 2}{5} \right) \right) = \frac{\sqrt{25 - (x + 2)^2}}{5}.
\]

### Step 5: Determining the Domain of \( f(x) \)

For the function \( \cos^{-1} \left( \frac{x + 2}{5} \right) \) to be valid, the expression inside the inverse cosine function must lie within the range \([-1, 1]\):

\[
-1 \leq \frac{x + 2}{5} \leq 1.
\]

Multiplying all parts of the inequality by 5:

\[
-5 \leq x + 2 \leq 5.
\]

Subtracting 2 from all parts:

\[
-7 \leq x \leq 3.
\]

Thus, the domain of \( f(x) \) is:

\[
\boxed{[-7, 3]}.
\]

### Final Simplified Expression and Domain

The simplified form of the function is:

\[
f(x) = \frac{\sqrt{25 - (x + 2)^2}}{5},
\]

with the domain \( x \in [-7, 3] \).

---

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### Related Questions:

1. How do you determine the range of the inverse trigonometric functions?
2. What are some applications of using trigonometric identities in solving equations?
3. How can we graph \( f(x) = \sin \left( \cos^{-1} \left( \frac{x+2}{5} \right) \right) \)?
4. What other functions could we simplify using a similar approach?
5. How does the Pythagorean theorem relate to trigonometric identities?

**Tip:** Always remember the domain restrictions when working with inverse trigonometric functions.

Use a right triangle to simplify f(x) = sin[cos^(-1)((x+2)/5)]. Determine the domain of f(x).

Learn how to simplify the trigonometric function f(x) = sin(cos^(-1)((x+2)/5)) using a right triangle. Explore the use of inverse trigonometric functions, the Pythagorean theorem, and domain restrictions. Suitable for students in Grades 11-12.

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