Sine and cosine are fundamental trigonometric functions. In a right triangle, sine of an angle theta equals the opposite side divided by the hypotenuse, while cosine equals the adjacent side divided by the hypotenuse.
On the unit circle, sine and cosine have a geometric interpretation. For any angle theta, sine equals the y-coordinate and cosine equals the x-coordinate of the point where the angle intersects the circle. This leads to the fundamental identity: sine squared plus cosine squared equals one.
The graphs of sine and cosine reveal their periodic nature. Sine starts at zero and reaches its maximum at pi over two, while cosine starts at one and reaches zero at pi over two. Both functions repeat every two pi units, making them periodic with period two pi.
Let's examine key angle values. At zero degrees, sine is zero and cosine is one. At thirty degrees, sine is one half and cosine is square root three over two. At forty-five degrees, both sine and cosine equal square root two over two. At sixty degrees, sine is square root three over two and cosine is one half. Finally, at ninety degrees, sine is one and cosine is zero.
To summarize what we have learned: Sine and cosine are fundamental trigonometric functions that relate angles to ratios in right triangles and coordinates on the unit circle. Both functions are periodic with period two pi and have range from negative one to positive one. The Pythagorean identity shows that sine squared plus cosine squared always equals one. These functions form the foundation of trigonometry and have countless applications in mathematics, physics, and engineering.