To find the adjacent side length of the triangle given in the image, we can use trigonometric functions.

In the given right triangle:

• The side opposite angle $\theta$ is 8.
• The hypotenuse is 15.
• We need to find the adjacent side.

Using the Pythagorean theorem:

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

Substituting the given values:

$15^2 = 8^2 + \text{Adjacent}^2$

$225 = 64 + \text{Adjacent}^2$

$\text{Adjacent}^2 = 225 - 64$

$\text{Adjacent}^2 = 161$

$\text{Adjacent} = \sqrt{161}$

The exact value of the adjacent side is $\sqrt{161}$.

Would you like more details or have any questions? Here are some related questions:

1. How can we find the exact value of $\sin(\theta)$ using the triangle?
2. What is the value of $\cos(\theta)$ using the adjacent side we found?
3. How does the Pythagorean theorem apply to other types of triangles?
4. What are other methods to find the adjacent side without using the Pythagorean theorem?
5. How can trigonometric identities be used to verify this result?

Tip: Always double-check your calculations, especially when dealing with square roots and algebraic manipulations to avoid mistakes!