To find the sine of the angle opposite the side $x$ in a right triangle, you can use the sine function definition:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Given:

• Hypotenuse = 9
• Adjacent side = 7
• Opposite side = $x$

Here, the sine of the angle $\theta$ opposite side $x$ is:

$\sin(\theta) = \frac{x}{9}$

Since the value of $x$ (the opposite side) is not given directly, we must calculate it using the Pythagorean theorem:

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

Substituting the given values:

$9^2 = 7^2 + x^2$

$81 = 49 + x^2$

Solve for $x^2$:

$x^2 = 81 - 49 = 32$

Take the square root of both sides:

$x = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$

Now, substitute the value of $x$ into the sine formula:

$\sin(\theta) = \frac{4\sqrt{2}}{9}$

So, the sine of the angle opposite the side $x$ is:

$\sin(\theta) = \frac{4\sqrt{2}}{9}$

Would you like to go over any specific details, or do you have any further questions?

Here are 8 related questions you might find interesting:

1. How do you calculate the cosine of $\theta$ in the same triangle?
2. What is the tangent of $\theta$ in this triangle?
3. How do you find the area of this right triangle?
4. Can you express the angle $\theta$ in degrees using the sine function?
5. What are the values of the sine, cosine, and tangent for the angle adjacent to side $x$?
6. How does the sine function change if the opposite side increases?
7. What is the length of the other angle in the triangle?
8. How do you calculate the perimeter of this triangle?

Tip: The Pythagorean theorem is a powerful tool for determining the missing side of a right triangle, and it can be especially useful in trigonometric calculations.