https://file.aiquickdraw.com/mathbot/file/1725469898472ngp96og4.jpg

It looks like you have uploaded a picture showing two triangles with angles and side lengths, where the goal is to find the value of \( x \). Let's solve both of them step by step.

### Problem 3:
- The triangle is a right triangle (indicated by the right angle).
- Given:
  - One angle is \( 30^\circ \).
  - The adjacent side is \( 63.1 \) units.

We can use the trigonometric function:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Where \( \theta = 30^\circ \), adjacent side = 63.1, and hypotenuse = \( x \).

\[
\cos(30^\circ) = \frac{63.1}{x}
\]
Now, solve for \( x \).

### Problem 4:
- Another triangle (not a right triangle).
- Given:
  - One angle is \( 25^\circ \).
  - One side opposite this angle is \( 19.2 \).

To solve this, we can use the law of sines:
\[
\frac{\sin(A)}{a} = \frac{\sin(B)}{b}
\]
Where:
  - \( A = 25^\circ \),
  - \( a = 19.2 \),
  - \( B = 90^\circ \), since the sum of angles in a triangle is 180°.

This can help us find side \( x \).

Let me calculate these values for you.Here are the values of \( x \) for both problems:

1. **Problem 3**: The hypotenuse of the triangle is approximately \( x = 72.9 \) units (rounded to the nearest tenth).
2. **Problem 4**: The side opposite the 90° angle is approximately \( x = 45.4 \) units (rounded to the nearest tenth).

Would you like further details on these calculations or have any questions?

Here are 5 related questions for deeper understanding:
1. How can the sine and cosine rules be used to solve other types of triangles?
2. What is the importance of rounding in trigonometric calculations?
3. How would the problem change if the given angle were different?
4. Could the same principles apply if the triangle were in a 3D space?
5. How do you identify when to use the law of sines versus the law of cosines?

**Tip**: Always ensure your calculator is in the correct mode (degrees or radians) depending on the angle measurements used in trigonometric functions.

Find the value of x. Round to the nearest tenth, if necessary. (Based on the two triangles provided in the image)

This problem involves finding the value of x in two triangles, one of which is a right triangle. Learn how to use trigonometric functions such as cosine and the law of sines to determine the side lengths. Perfect for students in Grades 9-10.

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