---
title: "Periodicity of sin 6πt, cos 8t and their sum"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:periodicity-sin6pit-cos8t"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodicity-sin6pit-cos8t
site: BCT Engineering
---

# Periodicity of sin 6πt, cos 8t and their sum

Signals and Systems · Chapter 1: Signals

- Asked in: 2077 Chaitra, 3 marks
- Video: [Find whether the following signals are periodic or not, If periodic find the fundamental period.](https://www.youtube.com/watch?v=9b-yKbDVNUo) — Engg-Course-Made-Easy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodicity-sin6pit-cos8t
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

x(t) and y(t) are continuous time signals defined as: x(t) = sin 6πt and y(t) = cos 8t. Determine whether: (i) x(t) is periodic, (ii) y(t) is periodic, (iii) x(t) + y(t) is periodic. If periodic, find the fundamental time period of x(t), y(t) and x(t) + y(t).

## Short answer

x(t)=sin6πt: T₁=2π/(6π)=1/3 s (periodic). y(t)=cos8t: T₂=2π/8=π/4 s (periodic). Their sum is not periodic because T₁/T₂ = (1/3)/(π/4) = 4/(3π) is irrational, so no common period exists.

## Answer

A CT sinusoid with angular frequency $\omega$ is always periodic with $T = 2\pi/\omega$. A sum is periodic only if $T_1/T_2$ is rational.

**(i)** $x(t) = \sin 6\pi t$, $\omega_1 = 6\pi$:

$$
T_1 = \frac{2\pi}{6\pi} = \frac{1}{3}\ \text{s} \quad\Rightarrow\quad \text{periodic}
$$

**(ii)** $y(t) = \cos 8t$, $\omega_2 = 8$:

$$
T_2 = \frac{2\pi}{8} = \frac{\pi}{4}\ \text{s} \approx 0.785\ \text{s} \quad\Rightarrow\quad \text{periodic}
$$

**(iii)** $x(t) + y(t)$:

$$
\frac{T_1}{T_2} = \frac{1/3}{\pi/4} = \frac{4}{3\pi} \quad\text{(irrational)}
$$

No integers $m, k$ give $mT_1 = kT_2$, so there is no common period.

**Answer:** $x(t)$ periodic, $T_1 = 1/3$ s; $y(t)$ periodic, $T_2 = \pi/4$ s; $x(t) + y(t)$ is **not periodic** (aperiodic).

## Check yourself

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

1. What is the fundamental period of x(t) = sin 6πt?
   A. 1/3 s
   B. 1/6 s
   C. 3 s
   D. π/3 s
   Answer: A. T = 2π/ω = 2π/6π = 1/3 s. (Key point: CT sinusoid period)

2. What is the fundamental period of y(t) = cos 8t?
   A. 1/8 s
   B. π/4 s
   C. 8 s
   D. 2π s
   Answer: B. T = 2π/8 = π/4 s, about 0.785 s. (Key point: Period of cos 8t)

3. Why is x(t) + y(t) not periodic?
   A. Because cos 8t has no period
   B. Because sin and cos cannot be added
   C. T1/T2 = 4/(3π) is irrational, so no integers m, k give mT1 = kT2
   D. Because T1 is smaller than T2
   Answer: C. A sum of periodic signals is periodic only if T1/T2 is rational; here the ratio contains π. (Key point: Sum periodicity test)

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