Mathematically, a sine wave and a cosine wave are very similar. They are both periodic functions that oscillate between a minimum and maximum value. The primary difference lies in their phase.
A sine wave (sin wave) is a mathematical function that describes a smooth, repetitive oscillation. It can be represented by the equation y = A * sin(ωt + φ), where A represents the amplitude, ω is the angular frequency, t is the time, and φ is the phase shift.
A cosine wave (cos wave) is also a periodic function that oscillates between a minimum and maximum value. It can be represented by the equation y = A * cos(ωt + φ), using the same variables as the sine wave equation.
The primary distinction between a sine wave and a cosine wave is the phase shift (φ) in their equations. In a sine wave, the initial value occurs at φ = 0, while in a cosine wave, the initial value occurs at φ = π/2 or 90 degrees. This phase shift affects the starting position of the wave.
In terms of their shape, both sine and cosine waves have a smooth, repetitive oscillatory pattern. They are symmetrical about the y-axis and have the same amplitude, period, and frequency.
The amplitude of a sine wave or cosine wave represents the maximum displacement from the equilibrium position. It determines the height of the wave.
The periodicity or period of a sine wave or cosine wave is the length of one complete cycle. It is the distance between two corresponding points in the wave, such as two adjacent crests or troughs. The period is inversely related to the frequency, which is the number of cycles per unit time.
Since sine and cosine functions are trigonometric functions, they have a periodicity of 2π radians or 360 degrees. This means that the wave repeats itself after every 2π radians or 360 degrees.
In summary, mathematically speaking, the primary difference between a sine wave and a cosine wave is the phase shift in their equations. The sine wave starts at zero phase (φ = 0), while the cosine wave starts at a phase shift of π/2 (90 degrees). Both waves share the same shape, amplitude, period, and frequency, with periodicity of 2π radians or 360 degrees.