Sine and cosine are fundamental trigonometric functions. In a right triangle, sine of an angle theta equals the opposite side divided by the hypotenuse. Cosine of angle theta equals the adjacent side divided by the hypotenuse. The key difference is that sine uses the opposite side, while cosine uses the adjacent side.
On the unit circle, sine and cosine have a geometric interpretation. For any angle theta measured from the positive x-axis, the point where the angle intersects the unit circle has coordinates cosine theta and sine theta. Cosine represents the x-coordinate, while sine represents the y-coordinate. As the angle changes, we can see how these values change continuously.
The graphs of sine and cosine reveal their key differences. Both are periodic waves with the same shape, but they are phase-shifted. Sine starts at zero when the angle is zero, rises to one at π/2, returns to zero at π, drops to negative one at 3π/2, and completes the cycle at 2π. Cosine starts at one when the angle is zero, and follows the same pattern but shifted left by π/2 radians. This means cosine is simply sine shifted horizontally.
Let's examine key values that highlight the relationship between sine and cosine. At zero degrees, sine is zero and cosine is one. At thirty degrees, sine is one-half while 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, the values flip: sine becomes square root of three over two and cosine becomes one-half. Finally, at ninety degrees, sine reaches one and cosine becomes zero. Notice the fundamental relationship: sine squared plus cosine squared always equals one, and complementary angles show that sine of an angle equals cosine of its complement.
To summarize the key differences between sine and cosine: In right triangles, sine uses the opposite side while cosine uses the adjacent side, both divided by the hypotenuse. On the unit circle, sine gives the y-coordinate and cosine gives the x-coordinate of any point. Their graphs are identical waves, but cosine is sine shifted left by π/2 radians. They satisfy the fundamental identity that sine squared plus cosine squared equals one. Understanding these complementary functions is essential for trigonometry, calculus, physics, and engineering applications.