To simplify the expression $f(x) = \sin(\cos^{-1}((x+2)/5))$, we can use a right triangle to represent the inverse trigonometric function and find the sine value. Here is the step-by-step approach:

Step 1: Understanding the Inverse Cosine

Given $y = \cos^{-1}((x+2)/5)$, it implies that: $\cos(y) = \frac{x+2}{5}$

This tells us that in a right triangle, the adjacent side to the angle $y$ is $x+2$ and the hypotenuse is $5$.

Step 2: Construct the Right Triangle

To find $\sin(y)$, we need the opposite side of the angle $y$. Using the Pythagorean theorem:

$\text{Opposite}^2 + \text{Adjacent}^2 = \text{Hypotenuse}^2$

Substitute the known values:

$\text{Opposite}^2 + (x+2)^2 = 5^2$

$\text{Opposite}^2 + (x+2)^2 = 25$

$\text{Opposite}^2 = 25 - (x+2)^2$

$\text{Opposite} = \sqrt{25 - (x+2)^2}$

Step 3: Find the Sine of the Angle

The sine of the angle $y$ is given by:

$\sin(y) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{\sqrt{25 - (x+2)^2}}{5}$

Step 4: Simplify $f(x)$

Now, substituting back into the function $f(x)$:

$f(x) = \sin(\cos^{-1}((x+2)/5)) = \frac{\sqrt{25 - (x+2)^2}}{5}$

Final Answer

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

This is the simplified form of the given function.

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Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:

1. How do you simplify other expressions involving inverse trigonometric functions using a right triangle?
2. What are some examples of trigonometric identities that involve inverse functions?
3. How can we determine the domain and range for the simplified function?
4. How would the function change if the expression inside the inverse cosine were different?
5. Can you solve for $x$ when $f(x) = 0$ in this case?

Tip: When dealing with inverse trigonometric functions, sketching a right triangle can often simplify the problem and help visualize relationships between the sides and angles.