Given the triangle 35x6045[Graphs generated by this script: initPicture(-15,55,-15,55);line([0,0],[25,50]);line([25,50],[50,0]);line([0,0],[50,0]);text([5,25],"35");text([45,25],"x");text([38,4],"60");text([8,4],"45");circle([13,7],1);circle([43,7],1)], find the length of side x using the Law of Sines.  Round your final answer to 2 decimal places.

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To solve for the length of side \( x \) in the given triangle using the Law of Sines, let's go through the process step by step.

### Step 1: Identify the given values and the Law of Sines
In this problem, we have:
- Side \( a = 60 \) opposite angle \( A = 35^\circ \)
- Side \( b = x \) opposite angle \( B = 45^\circ \)

The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

### Step 2: Apply the Law of Sines
Using the Law of Sines, we can write:
\[
\frac{60}{\sin(35^\circ)} = \frac{x}{\sin(45^\circ)}
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \):
\[
x = \frac{60 \times \sin(45^\circ)}{\sin(35^\circ)}
\]

Given:
\[
\sin(35^\circ) \approx 0.5736 \quad \text{and} \quad \sin(45^\circ) \approx 0.7071
\]

Substitute the values:
\[
x = \frac{60 \times 0.7071}{0.5736} \approx \frac{42.426}{0.5736} \approx 73.98
\]

### Final Answer
The length of side \( x \) is approximately \( \boxed{73.98} \).

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Would you like more details on how the Law of Sines is applied or have any other questions?

Here are some related questions for further practice:

1. How can the Law of Cosines be used to find the length of a side in a triangle?
2. How can you find the third angle in a triangle if two angles are known?
3. What steps should you take if you know all three angles but no sides?
4. How does the Law of Sines help in determining whether a triangle is right-angled?
5. Can the Law of Sines be applied to obtuse triangles? How?
6. What are the implications of rounding angles and sides in trigonometric calculations?
7. How do you solve a triangle if you know two angles and one side?
8. How does the Law of Sines change when using radians instead of degrees?

**Tip:** Always double-check the trigonometric values for accuracy, especially when rounding is involved, as it can significantly impact the final result.

Learn how to find the length of side x in a triangle using the Law of Sines with a detailed step-by-step solution. This problem involves trigonometric calculations with angles and side lengths, rounded to two decimal places. Suitable for high school students studying trigonometry.

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