---
title: "Periodicity of sin 5n and cos sum"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:periodicity-sin5n-cos-sum"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodicity-sin5n-cos-sum
site: BCT Engineering
---

# Periodicity of sin 5n and cos sum

Signals and Systems · Chapter 1: Signals

- Asked in: 2074 Bhadra, 4 marks
- Video: [Questions on periodicity || fundamental period of periodic signals](https://www.youtube.com/watch?v=QUCPlwiT7GI) — easy study plus
- 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-sin5n-cos-sum
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Determine whether the following discrete time signals are periodic or not (i) sin 5n (ii) cos(2πn/5) + cos(2πn/7)

## Short answer

For sin5n, ω₀=5 gives ω₀/(2π)=5/(2π), irrational, so it is not periodic. For cos(2πn/5)+cos(2πn/7), ω₁/(2π)=1/5 (N₁=5) and ω₂/(2π)=1/7 (N₂=7), both rational; the overall signal is periodic with fundamental period N = LCM(5,7) = 35.

## Answer

A DT sinusoid $\sin\omega_0 n$ is periodic only if $\omega_0/2\pi$ is rational; then $N = 2\pi m/\omega_0$ (smallest integer).

#### (i) $x[n] = \sin 5n$

$$
\frac{\omega_0}{2\pi} = \frac{5}{2\pi} \quad\text{(irrational)}
$$

No integer $N$ satisfies $5N = 2\pi m$. **Not periodic.**

#### (ii) $x[n] = \cos\left(\frac{2\pi n}{5}\right) + \cos\left(\frac{2\pi n}{7}\right)$

- Term 1: $\omega_1/2\pi = 1/5$, rational, $N_1 = 5$.
- Term 2: $\omega_2/2\pi = 1/7$, rational, $N_2 = 7$.

$$
N = \text{LCM}(5, 7) = 35
$$

**Answer:** (i) aperiodic; (ii) periodic with fundamental period $N = 35$ samples.

## Check yourself

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

1. Why is sin 5n not periodic?
   A. 5 is an odd number
   B. ω0/2π = 5/2π is irrational
   C. 5 is greater than 2π
   D. Sine is never periodic in DT
   Answer: B. No integer N makes 5N a multiple of 2π. (Key point: Irrational case)

2. What are the periods of cos(2πn/5) and cos(2πn/7)?
   A. 2π/5 and 2π/7
   B. 10 and 14
   C. 5 and 7
   D. 1/5 and 1/7
   Answer: C. ω/2π = 1/5 and 1/7, so N1 = 5 and N2 = 7. (Key point: Individual periods)

3. What is the fundamental period of the sum?
   A. 12
   B. 5
   C. 70
   D. 35
   Answer: D. N = LCM(5, 7) = 35. (Key point: Period of the sum)

4. When is a sum of DT periodic signals periodic?
   A. When each part is periodic, with period equal to the LCM of the periods
   B. Only when the periods are equal
   C. Only when one period divides the other
   D. Never
   Answer: A. Both components repeat after the LCM of their periods. (Key point: Sum rule)

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