# DSP - Sinewave: what is $2\pi f t$?

I am a beginner in DSP and is familiar with $\sin(\theta)$ declaration (as studied in trigonometry) - where $\theta$ is the angle it makes with X - axis. Please tell me what does $\sin(2\pi f t + \phi)$ equation means and how $\theta = 2\pi ft + \phi$.

A 1D sinusoidal wave, time dependent, is described by the next formula:

$$y(t) = A\sin(2\pi f t + \phi)$$

The trick here is just the concept. When you use the formula $A\sin(\theta)$ you aren't specifying if $\theta$ is time-dependent and you are not thinking in the formula as a wave, but as a mere function. But when you use the argument $2\pi f t$ you are considering the angular frequency $2\pi f$ ($rad/s$), where $f$ is the normal frequency, and the wave's phase $\phi$ (which represents a shift in the wave).

If you are working with a simple sine wave, you can assume a zero-shift, $\phi = 0$, and select a working frequency. Sample the wave and work normally with your DSP.

I hope it helps!

You can check sine wave description in WIKI

• Thanks I understand better know. Is there any book that explains these basic principles? It would really help me a lot. – Programmer Jul 3 '14 at 4:42
• Any book about DSP is pretty good, you only need a strong background in math and, to start, some notions about waves (signals). If you feel confident you can start reading this book. I found it pretty amazing for basic-intermediate level. Bests! – Fruzti Jul 3 '14 at 5:04
• Thanks for the link. It really helps me in my first step towards self-learning DSP – Programmer Jul 3 '14 at 13:39
• @Prakash 2πf = ω, which is angular frequency in radians/sec. So a simpler formula is $\sin(\omega t)$, which shows more clearly that you're just stretching or squishing the time axis of the sin wave to make it oscillate faster or slower. So the 2π is just a scaling factor from normal frequency to angular frequency. – endolith Jul 3 '14 at 14:21