Introduction to Trigonometry

Trigonometry is the study of triangles, focusing on the relationships between their angles and sides. Originating in ancient cultures, it's vital for navigation, astronomy, and solving complex geometrical problems.

A triangle has three sides and three angles. The sum of angles is always 180 degrees. Trigonometry uses sine, cosine, and tangent functions to relate angles with sides.

The sine of an angle is the ratio of the length of the opposite side to the hypotenuse. For angle θ, it's expressed as sin(θ) = opposite/hypotenuse.

Cosine relates the adjacent side to the hypotenuse, written as cos(θ) = adjacent/hypotenuse. Tangent is the opposite over adjacent, tan(θ) = opposite/adjacent. These functions are crucial for side calculations.

To find a missing side, identify the known sides/angles. Use the appropriate trigonometric function, and solve for the unknown variable. In right triangles, Pythagoras' theorem is also a valuable tool.

Surprisingly, trigonometry extends beyond triangles. It underpins Fourier transforms - crucial in signal processing and the decomposition of functions into oscillatory components, revolutionizing modern technology.

Trigonometry isn't just theoretical; it's practical in various fields. For instance, architects use it for creating structures, and it's essential in developing video games to simulate realistic movements and environments.