The graphs of sine and cosine have a lot in common with the swinging motion you see in a pendulum or a swing: they are both classic examples of repeating, backā€‘andā€‘forth motion over time.

Key things they have in common

  1. Repeating highs and lows
    • A swing goes up to a highest point, comes down through the middle, then reaches a lowest point on the other side, and repeats.
    • Sine and cosine graphs do the same: they have peaks (maximums) and valleys (minimums) that repeat in a regular pattern.
  2. Equal time for each cycle
    • If you push a swing with the same strength each time, each full backā€‘andā€‘forth swing takes about the same amount of time (its period).
    • On a sine or cosine graph, each full wave (from peak back to peak, or from any point back to the same point moving in the same direction) also takes the same horizontal distance, called the period.
  3. Smooth backā€‘andā€‘forth motion
    • The swing doesnā€™t suddenly stop and turn around; it slows down smoothly at the top, then reverses.
    • Sine and cosine graphs are smooth curves with no sharp corners, and at the top and bottom of each wave the graph ā€œturns aroundā€ gently, just like a swing changing direction.
  4. Middle position as the ā€˜restā€™ line
    • A swing passes fastest through its middle position (hanging straight down), which is its equilibrium or ā€œrestā€ position.
    • On a sine or cosine graph, this is the horizontal axis of the wave (the midline). The graph crosses this line in the middle of each swing from high to low.
  5. Same shape as simple harmonic motion
    • In physics, ideal swinging motion (for small angles) is called simple harmonic motion , and its position over time can be described using sine or cosine functions.
    • Thatā€™s why the swinging you see and the graphs of sine and cosine look so similar: theyā€™re actually using the same mathematical pattern.