The graphs of sine and cosine share several key features with the swinging motion you see in a pendulum or swing:

• Periodic motion : Both sine and cosine graphs are periodic functions, meaning they repeat their pattern over regular intervals. Their period is 2π2\pi 2π, which corresponds to one full cycle of swinging back and forth

• Smooth, wave-like shape : The swinging motion traces a smooth, continuous curve that rises and falls in a wave-like manner, just like the sine and cosine curves that oscillate between -1 and 1

• Amplitude corresponds to maximum displacement : The height of the sine or cosine wave (called amplitude) matches the maximum angular displacement of the swing from its resting position

• Symmetry and regularity : The swinging motion is symmetric in time-going forward and backward in a regular, repeating pattern-which is reflected in the symmetry of sine (odd function, symmetric about origin) and cosine (even function, symmetric about vertical axis) graphs

• Connection to circular motion : The sine and cosine functions arise from projecting uniform circular motion onto the vertical and horizontal axes. A swinging pendulum’s motion can be modeled as a projection of circular motion, which explains why its displacement over time follows sine or cosine curves

In summary, the swinging motion you see is a physical example of periodic oscillation that sine and cosine graphs mathematically describe. The smooth, repeating waveforms of these graphs mirror the back-and-forth rhythm of a swing or pendulum.