Sine and cosine are fundamental trigonometric functions that describe the relationship between angles and side lengths in right triangles. In a right triangle, we have an angle theta, the opposite side, the adjacent side, and the hypotenuse.
Sine of an angle theta is defined as the ratio of the opposite side to the hypotenuse. In mathematical notation, sine theta equals opposite divided by hypotenuse. This ratio tells us how the vertical component relates to the longest side of the triangle.
Cosine of an angle theta is defined as the ratio of the adjacent side to the hypotenuse. In mathematical notation, cosine theta equals adjacent divided by hypotenuse. This ratio tells us how the horizontal component relates to the longest side of the triangle.
On the unit circle, sine and cosine have a beautiful geometric interpretation. The sine of an angle is the y-coordinate of the point where the angle intersects the circle, while the cosine is the x-coordinate. As the angle changes, the point moves around the circle, and both sine and cosine values change smoothly.
To summarize: Sine equals the opposite side over the hypotenuse, while cosine equals the adjacent side over the hypotenuse. On the unit circle, sine gives us the y-coordinate and cosine gives us the x-coordinate. These fundamental trigonometric functions are essential tools for understanding relationships between angles and sides in triangles, and they have wide applications in mathematics, physics, and engineering.