Sine & Cosine Wave Calculator
Calculate the instantaneous value of a sine or cosine wave at any moment in time, given amplitude, frequency, and phase shift. Essential for AC electronics, audio, and signal processing work.
About this calculator
A sine wave is described by y(t) = A × sin(2π·f·t + φ) and a cosine wave by y(t) = A × cos(2π·f·t + φ), where A is the amplitude (peak value), f is the frequency in Hz, t is the time in seconds, and φ is the phase shift in radians (converted from degrees). The term 2π·f·t represents the angular position of the wave at time t, and adding φ shifts the wave left or right along the time axis. The period of the wave is T = 1/f. Amplitude controls the peak-to-trough range (total swing = 2A), while frequency controls how many complete cycles occur per second. These formulas are used in AC power analysis, audio synthesis, and physics simulations.
How to use
Suppose you have a sine wave with amplitude A = 5 V, frequency f = 2 Hz, and phase shift φ = 45°. You want the value at t = 0.1 s. First, convert phase: 45° × π/180 = 0.7854 rad. Then apply: y = 5 × sin(2π × 2 × 0.1 + 0.7854) = 5 × sin(1.2566 + 0.7854) = 5 × sin(2.0420) = 5 × 0.8988 ≈ 4.49 V. Enter A = 5, f = 2, phase = 45°, t = 0.1, and select Sine to get this result instantly.
Frequently asked questions
What does phase shift do to a sine or cosine wave in practice?
Phase shift moves the wave horizontally along the time axis. A positive phase shift displaces the wave to the left (it starts earlier), while a negative phase shift moves it to the right (it starts later). In AC circuits, phase shift between voltage and current determines whether a circuit is capacitive or inductive and directly affects real power delivery. In audio, phase differences between channels can cause cancellation or reinforcement of specific frequencies, affecting sound quality and stereo imaging.
How does frequency affect the shape and period of a sine wave?
Frequency (f, in Hz) determines how many complete cycles occur per second. The period T = 1/f is the time for one full cycle. Doubling the frequency halves the period, making the wave appear compressed along the time axis. In audio, higher frequency corresponds to higher pitch — a 440 Hz wave produces the musical note A4. In electrical engineering, mains power runs at 50 or 60 Hz depending on the country, and this frequency governs the design of transformers, motors, and power supplies.
What is the difference between a sine wave and a cosine wave and when should I use each?
A cosine wave is simply a sine wave shifted 90° (π/2 radians) to the left: cos(θ) = sin(θ + π/2). They represent the same oscillation but with a different starting phase. Sine waves are conventionally used when the wave starts at zero at t = 0, while cosine waves start at their peak value. In AC circuit analysis, voltage is often expressed as a cosine and current as a sine to conveniently represent their phase difference. In mathematics, both forms are interchangeable — the choice is usually one of convention or convenience.