Using trigonometry, calculate the area of a triangle with sides measuring 10 cm and 15 cm, and an included angle of 30 degrees.

To find the area of the triangle, you can use the formula: Area = 0.5 * a * b * sin(C), where a and b are the lengths of the sides, and C is the included angle. Plugging in the values: Area = 0.5 * 10 cm * 15 cm * sin(30°) = 0.5 * 10 * 15 * 0.5 = 37.5 cm². So, the area of the triangle is 37.5 cm²! Did you know that the concept of calculating areas using trigonometry dates back to ancient civilizations? The Greeks were particularly fascinated by geometry and used trigonometric principles in diverse applications, from astronomy to architecture. It's amazing to think that principles we use today were being explored thousands of years ago! Triangles are not just theoretical shapes; they play key roles in modern engineering! For example, trusses, which are used in bridges and roofs, are often designed using triangular formations. Their strength and stability come from the properties of triangles, which allow for the even distribution of weight. So, the next time you see a bridge, you can appreciate how trigonometry helps keep it standing strong!