---
title: "Is sin(t) periodic; energy or power"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:sine-periodic-energy-power"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/sine-periodic-energy-power
site: BCT Engineering
---

# Is sin(t) periodic; energy or power

Signals and Systems · Chapter 1: Signals

- Asked in: 2078 Poush, 1+3 marks
- Video: [Power of Sinusoidal Function 💡 Signal & System : Lec 23](https://www.youtube.com/watch?v=hTeWW7ZX-V0) — Mohan Genix
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/sine-periodic-energy-power
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Given: x(t) = sin(t). Is this signal periodic? Is it an energy signal or power signal or neither?

## Short answer

x(t)=sin t is periodic with fundamental period 2π s. Its energy over infinite time is infinite, and its average power is P=0.5 W (normalized), so it is a power signal.

## Answer

#### Periodicity
$\sin(t+T) = \sin t$ for all $t$ when $T = 2\pi m$. The smallest positive value is

$$
T_0 = \frac{2\pi}{\omega_0} = \frac{2\pi}{1} = 2\pi\ \text{s}
$$

So **$x(t) = \sin t$ is periodic** with $T_0 = 2\pi$.

#### Energy or power
**Energy:** $E = \int_{-\infty}^{\infty}\sin^2 t\,dt = \infty$ (the area of each period adds up forever).

**Power** (average over one period):

$$
\begin{aligned}
P &= \frac{1}{2\pi}\int_0^{2\pi}\sin^2 t\,dt = \frac{1}{2\pi}\int_0^{2\pi}\frac{1-\cos 2t}{2}\,dt \\
&= \frac{1}{2\pi}\cdot\frac{2\pi}{2} = \frac{1}{2}
\end{aligned}
$$

**Answer:** $\sin t$ is periodic ($T_0 = 2\pi$ s) and is a **power signal** with $P = 0.5$ W (normalised), $E = \infty$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The fundamental period of sin t is:
   A. pi
   B. 2 pi
   C. 1
   D. 4 pi
   Answer: B. T0 = 2 pi/w0 with w0 = 1 gives 2 pi. (Key point: Period of sin t)

2. The total energy of sin t over all time is:
   A. pi
   B. 1/2
   C. Infinite
   D. 0
   Answer: C. Each period adds the same non-zero energy, and there are infinitely many periods. (Key point: Energy)

3. The average power of sin t is:
   A. 1
   B. 1/2
   C. pi
   D. 2 pi
   Answer: B. (1/2pi) integral of sin^2 over a period = 1/2. (Key point: Power value)

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