To find the exact value of $y$, we are given the equation:

$y = \sin\left( \tan^{-1}(5) \right)$

Here’s the step-by-step process:

Step 1: Analyze $\tan^{-1}(5)$

The expression $\tan^{-1}(5)$ represents the angle $\theta$ whose tangent is 5. In other words:

$\tan(\theta) = 5$

We can think of this in terms of a right triangle, where the opposite side is 5, and the adjacent side is 1 (since tangent is opposite over adjacent). Using the Pythagorean theorem, we can find the hypotenuse:

$\text{Hypotenuse} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}$

Step 2: Find $\sin(\theta)$

Now, $\sin(\theta)$ is the ratio of the opposite side to the hypotenuse. Thus:

$\sin(\theta) = \frac{5}{\sqrt{26}} = \frac{5}{\sqrt{26}} \times \frac{\sqrt{26}}{\sqrt{26}} = \frac{5\sqrt{26}}{26}$

Final Answer:

The exact value of $y$ is:

$y = \frac{5\sqrt{26}}{26}$

Would you like more details or further explanation?

Here are 5 related questions to expand on this:

1. How do you compute the inverse tangent of other numbers?
2. Can the sine of an angle exceed 1 or be less than -1?
3. How do you compute the value of $\tan^{-1}$ for more complicated numbers?
4. What is the significance of the Pythagorean theorem in trigonometric identities?
5. How do inverse trigonometric functions relate to their respective trigonometric functions?

Tip: Understanding trigonometric identities helps in simplifying and solving problems involving inverse functions.