Wow, I spent a long time looking for links. Oh, well, I'll leave my answer after
frankvan's more concise one anyway.
Trigonometry is used for measuring triangles (surveying, architecture, engineering, navigation), sound wave analysis (music acoustics, voice recognition software), optics, physics (ballistics, ossilating systems), astronomy (calculating the distance to stars), computer/electrical uses (circuitry), and as a stepping stone to calculus used in many more areas.
From
http://standards.nctm.org/Previous/CurrEvStds/9-12s9.htm:
"Trigonometry has its origins in the study of triangle measurement. Many real-world problems, including those from the fields of navigation and surveying, require the solution of triangles. In addition, important mathematical topics, such as matrix representations of rotations, direction angles of vectors, polar coordinates, and trigonometric representations of complex numbers, require trigonometric ratios, further underscoring the connections between geometry and algebra.
"Natural generalizations of the ratios of right-angle trigonometry give rise to both trigonometric and circular functions. These functions, especially the sine and cosine, are mathematical models for many periodic real-world phenomena, such as uniform circular motion, temperature changes, biorhythms, sound waves, and tide variations."