Phase Shift of Sine and Cosine Waves

Review y = a*sin(b*x) y = a*cos(b*x) a  amplitude Vertical stretch/shrink b  frequency Horizontal stretch/shrink

This time y = a*sin(b*x+c) y = a*cos(b*x+c) What is c?

y = a*sin(b*x+c) c is called the phase angle, and it affects horizontal shift or displacement of the graph

Phase shift y = a*sin(b*x+c) c  phase angle Phase shift = -c/b Displacement graph is shifted by Note that phase angle and phase shift are not the same thing

Phase shift y =a*sin(b*x+c) y = sin(x+π) Phase shift = -c/b = -π/1 = -π y = sin(3x-π) Phase shift = -c/b = π/3 y =12*sin(2*x+π/4) Phase shift = -c/b = (-π/4)/2 = -π/8

Phase Shift Consider y = sin(x) and y = sin(x + π/4) Phase shift = -c/b = -π/4

Phase Shift Consider y = sin(x) and y = sin(x – π/2) Phase shift = -c/b = π/2

Phase Shift Notice: y = a* sin(b*x+c) shift by c/b in negative direction y = a* sin(b*x-c) shift by c/b in positive direction

Why do we care? Applications of sine waves in sciences, medical fields, engineering National Academy of Engineering Grand Challenge- Reverse-Engineer the Brain Neurons in the brain have electrical activity even after paralysis This activity can be recorded, and often is periodic like sine waves

Why do we care? Complex signals can be broken down into simpler sine waves Pattern recognition methods applied to find motor commands within the recorded signal Motor commands can be translated to robotic arms