The sine and cosine functions are fundamental trigonometric functions that relate angles to ratios in right triangles. In a right triangle, we have angle theta, the opposite side b, the adjacent side a, and the hypotenuse c.
The sine function is defined as the ratio of the opposite side to the hypotenuse. In our triangle, sine theta equals b divided by c, where b is the side opposite to angle theta, and c is the hypotenuse.
The cosine function is defined as the ratio of the adjacent side to the hypotenuse. In our triangle, cosine theta equals a divided by c, where a is the side adjacent to angle theta, and c is the hypotenuse.
On the unit circle, sine and cosine have a geometric interpretation. For any angle theta, the sine equals the y-coordinate of the point on the circle, while the cosine equals the x-coordinate. Both functions have a range from negative one to positive one.
The graphs of sine and cosine functions show their periodic nature. Both functions oscillate between negative one and positive one with a period of two pi. The sine function starts at zero, while the cosine function starts at one. These functions have applications in physics, engineering, and many other fields.