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To solve for \( x \) in the given triangle, we can use the Law of Cosines. The Law of Cosines states:

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Here, \( a \), \( b \), and \( c \) are the sides of the triangle, and \( C \) is the angle opposite side \( c \). In the given triangle:
- \( a = 6 \)
- \( b = 8 \)
- \( C = 52^\circ \)
- \( c = x \)

Plugging these values into the Law of Cosines formula, we get:

\[ x^2 = 6^2 + 8^2 - 2 \cdot 6 \cdot 8 \cdot \cos(52^\circ) \]

First, calculate the cosine of \( 52^\circ \):

\[ \cos(52^\circ) \approx 0.6157 \]

Next, substitute this value into the equation:

\[ x^2 = 6^2 + 8^2 - 2 \cdot 6 \cdot 8 \cdot 0.6157 \]
\[ x^2 = 36 + 64 - 96 \cdot 0.6157 \]
\[ x^2 = 100 - 59.1072 \]
\[ x^2 \approx 40.8928 \]

Finally, take the square root of both sides to solve for \( x \):

\[ x \approx \sqrt{40.8928} \]
\[ x \approx 6.4 \]

Therefore, the length of side \( x \) is approximately \( 6.4 \) units.

Would you like more details or have any questions about this solution?

Here are 5 more questions you might find interesting:
1. How do you use the Law of Sines in a triangle?
2. What is the Pythagorean Theorem and how is it used in right triangles?
3. How do you find the area of a triangle given its sides and an angle?
4. What are the different types of triangles and their properties?
5. How do you solve a system of linear equations using matrices?

**Tip:** When working with trigonometric functions, it's important to ensure your calculator is set to the correct mode (degrees or radians) based on the angle measure you are using.

Learn how to solve for x in a triangle using the Law of Cosines with a detailed step-by-step solution. This problem involves trigonometric concepts, including the Law of Cosines formula applied to find the length of a triangle side. Suitable for high school students.

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