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Finding the area of a triangle is a fundamental concept in geometry. There are various formulas to determine the area, but when dealing with triangles having known side lengths and the angle between them, the formula changes. In this case, we use the formula involving the sine of the angle:

\[ Area = \frac{1}{2} \times a \times b \times \sin(C) \]

Here, 'a' and 'b' are the lengths of the two sides, and 'C' is the angle between those sides. The sine function helps in accounting for how the angle affects the area. This formula is especially useful for non-right triangles, where traditional base-height methods aren't directly applicable.

Remember:

• The sides and angles must be in the same unit (e.g., angles in radians, sides in meters).
• This formula works well when you have side-angle-side (SAS) information.

It's a straightforward method that neatly handles various triangle configurations.

The sine function is one of the basic trigonometric functions. It helps relate the angles of a triangle to the ratios of its sides. For an angle in a right triangle, the sine is the ratio of the length of the opposite side to the hypotenuse.

In non-right triangles, the sine function's role extends, particularly in the law of sines and area calculations. The sine of special angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\) are typically memorized due to their frequent use.

For an angle of \(60^\circ\), as in our example, the sine value is:

• \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

This value is key to determining the area in our problem. The sine function converts the geometric problem into an algebraic computation, simplifying the solution process.

Converting angles from degrees to radians is essential in trigonometry, especially when using trigonometric functions. Most formulas in higher mathematics assume angle measurements in radians. To convert an angle from degrees to radians, we use the conversion factor:

\[1 \text{ radian} = \frac{180^\circ}{\pi}\]

For converting \(60^\circ\) to radians, you multiply:

• \(60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \text{ radians} \)

This conversion is crucial as it allows us to use the sine function correctly in the area formula. Always check to ensure that your angles are in radians when working with trigonometric functions in mathematical equations.