Dynamic Demonstration of Trigonometric Induction Formula I (Periodicity)

This applet dynamically demonstrates Trigonometric Induction Formula I, illustrating the periodic relationships of trigonometric functions, namely sin(α+2kπ)=sinα, cos(α+2kπ)=cosα, and tan(α+2kπ)=tanα. By constructing the terminal sides of angle α and its sum with 2kπ on the unit circle, the applet visually reveals how the coordinates of the corresponding points remain unchanged, thereby providing an intuitive geometric explanation for the periodic nature of sine, cosine, and tangent functions. Users can adjust the angle α and the integer parameter k through interactive sliders, observing in real time that rotating the terminal side by full circles (multiples of 2π) returns the point to the same position on the unit circle. This hands-on exploration reinforces the key insight that trigonometric function values depend only on the terminal side's direction, not on how many complete rotations precede it. The applet is designed for high school mathematics instruction and review in the trigonometry unit, offering teachers a powerful visual tool to supplement traditional lecture-based lessons and helping students transition from abstract formula memorization to genuine conceptual understanding.