Sine and cosine are fundamental trigonometric functions that describe relationships between angles and sides in right triangles. 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. These ratios extend beyond triangles to the unit circle, where sine represents the y-coordinate and cosine represents the x-coordinate of a point on the circle.
On the unit circle, sine and cosine take on a more general meaning. For any angle theta, we can find a corresponding point on the unit circle. The x-coordinate of this point is the cosine value, and the y-coordinate is the sine value. This definition works for all angles, not just those in right triangles. Let's observe how these values change as we rotate around the circle, visiting key angles like 30, 45, 60, and 90 degrees.
The graphs of sine and cosine functions reveal their wave-like nature and important properties. Both functions have an amplitude of 1, meaning they oscillate between -1 and 1. They have a period of 2π, completing one full cycle every 2π units. The domain extends to all real numbers, while the range is bounded between -1 and 1. A key relationship is that cosine leads sine by π/2 radians, which means cosine equals sine shifted left by π/2. We can visualize this connection by watching how the unit circle coordinates translate to the wave functions.
Let's analyze the key differences between sine and cosine functions. First, their starting points differ: sine starts at zero when x equals zero, while cosine starts at one. This reflects their different symmetry properties: sine is an odd function, meaning sine of negative x equals negative sine of x, while cosine is an even function, where cosine of negative x equals cosine of x. Their zeros occur at different intervals: sine equals zero at multiples of π, while cosine equals zero at π/2 plus multiples of π. These differences make each function suitable for different applications in mathematics and physics.
Sine and cosine functions have numerous practical applications in science and engineering. In simple harmonic motion, such as a pendulum, sine describes the displacement over time, starting from equilibrium. In AC electrical circuits, voltage is often represented by cosine, starting at maximum value, while current may use sine, starting at zero. Wave interference demonstrates how sine and cosine waves combine to create complex patterns.