Sine and cosine are fundamental trigonometric functions that relate angles to side ratios in right triangles. In a right triangle, sine of an angle theta equals the opposite side divided by the hypotenuse, while cosine of theta equals the adjacent side divided by the hypotenuse.
On the unit circle with radius one, we can define sine and cosine more generally for any angle. The cosine of theta is the x-coordinate of the point where the angle intersects the circle, and the sine of theta is the y-coordinate. This definition works for all angles, not just those in right triangles.
The graphs of sine and cosine reveal their periodic nature. Both functions repeat every two pi radians and oscillate between negative one and positive one. The sine function starts at zero and reaches its maximum at pi over two, while the cosine function starts at one and reaches zero at pi over two. These wave-like patterns are fundamental in describing oscillatory phenomena.
There are several important angle values that are commonly used. At zero degrees, sine is zero and cosine is one. At thirty degrees, sine is one half and cosine is square root of three over two. At forty five degrees, both sine and cosine equal square root of two over two. At sixty degrees, sine is square root of three over two and cosine is one half. At ninety degrees, sine is one and cosine is zero. These values satisfy the fundamental identity: sine squared theta plus cosine squared theta equals one.
To summarize what we have learned about sine and cosine: These fundamental trigonometric functions relate angles to side ratios in right triangles. On the unit circle, cosine represents the x-coordinate and sine represents the y-coordinate. Both functions are periodic with period two pi and oscillate between negative one and positive one. They satisfy the fundamental identity sine squared theta plus cosine squared theta equals one, and are essential tools in physics, engineering, and mathematics.