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 $x \in [-7, 3]$.

Would you like more details on any part of this solution?

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.