---
title: "Periodicity of sin(15πt) and sin(√2πt)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:periodic-check-sinusoids"
source: https://www.bctengineer.com/q/semester_5/data_communication/periodic-check-sinusoids
site: BCT Engineering
---

# Periodicity of sin(15πt) and sin(√2πt)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2075 Asoj, 3+3 marks
- Video: [Periodic and Non-Periodic Signals](https://www.youtube.com/watch?v=Gj9pDC_OBQU) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/periodic-check-sinusoids
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Determine whether the following signals are periodic or not: a) x(t) = sin 15πt  b) x(t) = sin √2πt

## Short answer

In continuous time, sin(ωt) is always periodic with period T=2π/ω. (a) sin(15πt) has T=2/15 s; (b) sin(√2·πt) has T=√2 s. Both signals are periodic.

## Answer

A continuous-time sinusoid sin(ωt) is periodic with period T = 2π/ω for any real ω ≠ 0.

#### a) x(t) = sin 15πt
ω = 15π rad/s, so T = 2π / 15π = 2/15 s ≈ 0.133 s.

**Periodic**, with fundamental period T = 2/15 s.

#### b) x(t) = sin √2·πt
Reading the argument as √2·π·t: ω = √2 π rad/s,

T = 2π / (√2 π) = 2/√2 = √2 s ≈ 1.414 s.

**Periodic**, with T = √2 s. The period is irrational, but that does not matter in continuous time: the condition for periodicity is only that a positive T exists such that x(t + T) = x(t).

(Note: if the signal were discrete-time, sin(√2 π n), it would be non-periodic, because 2π/ω₀ = √2 is not a rational number. For the continuous-time signal in the question, both signals are periodic.)

## Check yourself

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

1. sin(wt) period is
   A. w/2pi
   B. pi/w
   C. w
   D. 2pi/w
   Answer: D. Any real w. (Key point: Period)

2. sin15pi t has T
   A. 2/15 s
   B. 15 s
   C. 1/15 s
   D. 2 s
   Answer: A. 2pi/15pi. (Key point: Part a)

3. sin(sqrt2 pi t) is
   A. non-periodic
   B. periodic, T = sqrt2 s
   C. T = 2 s
   D. T = pi
   Answer: B. Irrational period fine in CT. (Key point: Part b)

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