,longest side of a triangle is 10 and the base is 8, find the area of the triangle if the angle between these two sides is 60 degrees. Please provide a complete solution step-by-step, not just the result, and use markdown format to make your steps clear. Also, the triangle area formula using sine should be explained.
user
Certainly! Below is the step-by-step solution to find the area of the triangle using the given information, and the explanation of the triangle area formula using sine is included.

### Problem Statement
Given:
- The longest side of the triangle \( a = 10 \)
- The base \( b = 8 \)
- The angle between these two sides \( \theta = 60^\circ \)

We need to find the area of the triangle.

### Step-by-Step Solution

1. **Understand the Triangle Area Formula Using Sine:**
   The area \( A \) of a triangle can be calculated using the formula:
   \[
   A = \frac{1}{2} \times a \times b \times \sin(\theta)
   \]
   where:
   - \( a \) and \( b \) are the lengths of two sides of the triangle.
   - \( \theta \) is the included angle between these two sides.

2. **Substitute the Given Values:**
   In this problem:
   - \( a = 10 \)
   - \( b = 8 \)
   - \( \theta = 60^\circ \)

   Using the sine of \( 60^\circ \):
   \[
   \sin(60^\circ) = \frac{\sqrt{3}}{2}
   \]

3. **Calculate the Area:**
   Substitute the values into the formula:
   \[
   A = \frac{1}{2} \times 10 \times 8 \times \sin(60^\circ)
   \]
   \[
   A = \frac{1}{2} \times 10 \times 8 \times \frac{\sqrt{3}}{2}
   \]

4. **Simplify the Expression:**
   \[
   A = \frac{1}{2} \times 10 \times 8 \times \frac{\sqrt{3}}{2}
   \]
   \[
   A = \frac{1}{2} \times 10 \times 4 \times \sqrt{3}
   \]
   \[
   A = 5 \times 4 \times \sqrt{3}
   \]
   \[
   A = 20 \sqrt{3}
   \]

### Final Answer
The area of the triangle is:
\[
\boxed{20\sqrt{3}}
\]

### Explanation of the Formula
The formula \( A = \frac{1}{2} \times a \times b \times \sin(\theta) \) is derived from the general formula for the area of a triangle, which is \( A = \frac{1}{2} \times \text{base} \times \text{height} \). In this case, the "height" is found using the sine of the included angle. The sine function gives the ratio of the opposite side to the hypotenuse in a right-angled triangle, which in this context, helps to find the perpendicular height from one side to the other.
assistant