1. Electronic communication
Most radio communication is based on the use of combinations of sines and cosine waves.
2. Thermal analysis
The heat equation is used to model how things get hot (electronics, spacecraft, ovens, etc). This equation is usually solved using sums of sines and cosines.
3. Signal Processing
The whole area of digital signal processing, which is used for HDTV and digital audio, is based on using sums of sines and cosines.
Earthquakes are modeled using the wave equation, which is frequently solved using sums of sines and cosines.
5. Building design
Buildings need to be designed to resist wind and earthquakes. The effect of waves on buildings is often modeled using sines and cosines to simulate wind and earth motion. These simulations determine how the buildings oscillate, which is also modeled by sines and cosines.