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In this problem, we are given a right triangle, and we need to find the length of side \(x\) using trigonometry. Here's what we know from the triangle:

- One angle is \(41^\circ\).
- The length of the opposite side (to the \(41^\circ\) angle) is 9.
- The adjacent side is labeled as \(x\), which we are trying to find.

Since we are dealing with a right triangle, we can use the tangent function:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substitute the known values into the equation:

\[
\tan(41^\circ) = \frac{9}{x}
\]

Now, solve for \(x\):

\[
x = \frac{9}{\tan(41^\circ)}
\]

I'll calculate this value now.The length of side \(x\) is approximately \(10.35\) units when rounded to the nearest hundredth.

Would you like further details or explanations on how this was calculated?

Here are 5 related questions to explore:
1. How can you use sine or cosine to find the hypotenuse in this triangle?
2. What is the relationship between the angle and the ratio of the sides in a right triangle?
3. How would the solution change if the given angle was larger, like 60°?
4. Can you find the hypotenuse using the Pythagorean theorem with the known values?
5. How would you apply this method to solve for a missing angle?

**Tip**: When dealing with right triangles, always decide which trigonometric function to use (sine, cosine, or tangent) based on which sides you know and which side you're trying to find.

For the right triangle below, find the length of x. Round to the hundredths. (2 decimal places)

Learn how to find the length of the adjacent side in a right triangle using the tangent function, given a 41° angle and an opposite side of 9 units. This trigonometry problem is suitable for students in Grades 9-10.

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