---
title: "Period of 3sin(7πn/5 + π/2)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:fundamental-period-3sin"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/fundamental-period-3sin
site: BCT Engineering
---

# Period of 3sin(7πn/5 + π/2)

Signals and Systems · Chapter 1: Signals

- Asked in: 2073 Bhadra, 4 marks
- Video: [Periodic Discrete Time Signals (Solved Problems)](https://www.youtube.com/watch?v=dD_67ZmfPTc) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/fundamental-period-3sin
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Calculate the fundamental period and fundamental frequency of the periodic signal x[n] = 3 sin((7π/5)n + π/2)

## Answer

$x[n] = 3\sin\left(\frac{7\pi}{5}n + \frac{\pi}{2}\right)$, with $\omega_0 = \frac{7\pi}{5}$.

**Check periodicity:**

$$
\frac{\omega_0}{2\pi} = \frac{7\pi/5}{2\pi} = \frac{7}{10} \quad\text{(rational, so periodic)}
$$

**Fundamental period:**

$$
N = \frac{2\pi m}{\omega_0} = \frac{2\pi m}{7\pi/5} = \frac{10m}{7}
$$

The smallest integer $N$ is obtained with $m = 7$:

$$
N = 10 \text{ samples}
$$

**Fundamental frequency:**

$$
\omega = \frac{2\pi}{N} = \frac{2\pi}{10} = \frac{\pi}{5}\ \text{rad/sample}, \qquad f = \frac{1}{N} = \frac{1}{10} = 0.1\ \text{cycles/sample}
$$

**Check:** $x[n+10] = 3\sin\left(\frac{7\pi}{5}n + 14\pi + \frac{\pi}{2}\right) = x[n]$ since $14\pi$ is a multiple of $2\pi$. The phase $\pi/2$ does not affect the period.

**Answer:** $N = 10$, fundamental frequency $\pi/5$ rad/sample (0.1 cycles/sample).

## Check yourself

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

1. What is ω0/2π for x[n] = 3 sin((7π/5)n + π/2)?
   A. 7/5
   B. 7/10
   C. 5/7
   D. Irrational
   Answer: B. ω0 = 7π/5, so ω0/2π = 7/10. (Key point: Rational test)

2. What is the fundamental period?
   A. 5 samples
   B. 7 samples
   C. 10 samples
   D. 70 samples
   Answer: C. N = 10m/7 and the smallest integer N occurs at m = 7, giving N = 10. (Key point: Fundamental period)

3. How does the phase π/2 affect the period?
   A. It doubles it
   B. It halves it
   C. It makes the signal aperiodic
   D. It does not affect it
   Answer: D. A constant phase shift only moves the waveform; the period depends on ω0. (Key point: Phase effect)

4. What is the fundamental frequency in rad/sample?
   A. π/5
   B. 7π/5
   C. 2π/5
   D. π/10
   Answer: A. ω = 2π/N = 2π/10 = π/5, equivalently 0.1 cycles/sample. (Key point: Fundamental frequency)

## Related in Signals and Systems

- [Period of 3sin(7πn/9 − 2) + cos(πn/13 + 2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afundamental-period-two-sinusoids) — Signals and Systems, chapter 1, asked 3×
- [Period of cos(πn/5)·sin(πn/3)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-period-cos-sin-product) — Signals and Systems, chapter 1
- [DTFT is periodic with period 2π](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-periodicity-2pi) — Signals and Systems, chapter 3
- [FT of periodic signals; e^(−2|t|)sin t](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-damped-sine) — Signals and Systems, chapter 3, asked 2×
- [Is sin(t) periodic; energy or power](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asine-periodic-energy-power) — Signals and Systems, chapter 1
- [Periodic signal whose Fourier series fails](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afourier-series-existence-examples) — Signals and Systems, chapter 2

## Related in other subjects

- [Periodic signals and fundamental period](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aperiodic-signals-fundamental-period) — Digital Signal Analysis and Processing, chapter 1
- [Angle modulated signal 12 sin(10⁶t + 5 sin10⁴t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-12sin-numerical) — Communication Systems, chapter 3
- [Fundamental frequency of a shaft: Rayleigh and Dunkerley](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Ashaft-rayleigh-dunkerley-numerical) — Theory of Mechanism and Machine II, chapter 9
- [Cycle, period, frequency, average and rms](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Adefine-cycle-period-frequency-avg-rms) — Basic Electrical Engineering, chapter 5