Find the exact value of y or state that y is undefined.
yequalssine left parenthesis tangent Superscript negative 1 Baseline 5 right parenthesis


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.

Find the exact value of y or state that y is undefined.
\( y = \sin( \tan^{-1}(5) ) \)

Find the exact value of y in the equation \( y = \sin( \tan^{-1}(5) ) \) using trigonometric concepts. This solution covers the use of the Pythagorean theorem to determine the sine of an angle from its inverse tangent. Suitable for students in Grades 10-12.

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