Great question! Trigonometric functions are derived from the relationships between the sides and angles of right triangles. The three main trigonometric functions - sine, cosine, and tangent - are defined as follows:

Sine (sin θ) = Opposite side / Hypotenuse

Cosine (cos θ) = Adjacent side / Hypotenuse

Tangent (tan θ) = Opposite side / Adjacent side

These ratios were originally developed to solve problems involving right triangles, where the goal was to find unknown side lengths or angle measures given certain known information.

The key problem that trigonometry is used to solve is finding missing information in right triangles. Some common applications include:

Surveying and navigation - Determining distances, heights, and angles in the real world using trigonometric principles.

Engineering and construction - Calculating load-bearing capacities, designing structures, and laying out foundations.

Astronomy - Measuring distances to celestial bodies, calculating orbits, and predicting the motion of planets and stars.

Physics - Analyzing wave phenomena, modeling projectile motion, and understanding electrical circuits.

The ability to use trigonometric functions to solve for unknown values in right triangles has made trigonometry an invaluable tool across many scientific and mathematical disciplines. The foundational concepts were developed centuries ago, but their applications continue to evolve and expand.