The Natural Cosine and Sine Curves

Bob Palais


This mathlet illustrates the relationship between the graphs of the natural (radian measure) sine and cosine curves and the vertical and horizontal coordinates of points on the unit circle. It can be used in calculus classes to provide an alternative way of understanding the derivative rules for sine and cosine. (See the text frame for more details.)

Bob Palais is a Research Associate Professor of Mathematics at the University of Utah.


  • Class demo
  • Student use


Browsers: Internet Explorer 5, Netscape 4,

Plugins: Flash,

Operating Systems: Mac OS 8 or 9, Windows 95, Windows 98, Windows NT


This applet is offered in the spirit of animating proofs without words. We hope to convey the meaning of the natural sine and cosine graphs and the symbolic formulas for their periodicity, phase shifts, rates of change can be understood dynamically, visually, in unison. It also serves as the basis for a unified rotational approach to the addition formulas, complex multiplication, and the geometric interpretation of the dot product. It might be most helpful in looking ahead to preview or looking back to review the connections among the different ways of viewing these seemingly diverse concepts.

Open The Natural Sine and Cosine Curves in a new window.


Grateful thanks to Tom Roby, David A. Smith, Tom Farmer, and Nick Korevaar for their encouragement and contributions to the presentation, and to Dale Meier for sharing his animation and design expertise.

Published December, 2001
© 2001 by Bob Palais