---
title: "Energy or power: 3cos(4πt), sin(n/6)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-3cos-sin"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-3cos-sin
site: BCT Engineering
---

# Energy or power: 3cos(4πt), sin(n/6)

Signals and Systems · Chapter 1: Signals

- Asked in: 2083 Baisakh (new course), 4 marks
- Video: [Energy and Power Signals | Solved Problems / Examples](https://www.youtube.com/watch?v=bZgQD_7d5VM) — Shrenik Jain
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-3cos-sin
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Determine whether the following signal is an energy or power signal. (i) f(t) = 3cos(4πt) (ii) x[n] = sin(n/6)

## Short answer

f(t)=3cos(4πt) has infinite energy and average power 4.5 W, so it is a power signal. x[n]=sin(n/6) also has infinite energy and 0.5 W average power, thus a power signal.

## Answer

A signal with finite non-zero energy is an energy signal; one with finite non-zero average power (and infinite energy) is a power signal.

#### (i) $f(t) = 3\cos(4\pi t)$
Periodic with $T = 2\pi/4\pi = 0.5$ s, so $E = \infty$.

$$
\begin{aligned}
P &= \frac{1}{T}\int_0^{T}9\cos^2(4\pi t)\,dt = \frac{1}{0.5}\int_0^{0.5}\frac{9}{2}\left[1 + \cos(8\pi t)\right]dt \\
&= 2\times\frac{9}{2}\times 0.5 + 0 = 4.5\ \text{W}
\end{aligned}
$$

**Answer:** $E = \infty$, $P = 4.5$ W, so $f(t)$ is a **power signal**.

#### (ii) $x[n] = \sin(n/6)$
$\omega_0/2\pi = 1/(12\pi)$ is irrational, so $x[n]$ is aperiodic; its samples never die out, so $E = \sum \sin^2(n/6) = \infty$.

$$
\begin{aligned}
P &= \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\sin^2\frac{n}{6} \\
&= \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\frac{1}{2}\left[1 - \cos\frac{n}{3}\right] = \frac{1}{2} - 0 = 0.5\ \text{W}
\end{aligned}
$$

(the sum of $\cos(n/3)$ is bounded, so after dividing by $2N+1$ it tends to 0).

**Answer:** $E = \infty$, $P = 0.5$ W, so $x[n]$ is a **power signal**.

## Check yourself

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

1. What is the period of 3 cos(4πt)?
   A. 2 s
   B. 1 s
   C. 0.5 s
   D. 4π s
   Answer: C. T = 2π/4π = 0.5 s. (Key point: Period of 3cos(4πt))

2. What is its average power?
   A. 9 W
   B. 3 W
   C. 2.25 W
   D. 4.5 W
   Answer: D. P = A^2/2 = 4.5 W, independent of frequency. (Key point: Power of 3cos(4πt))

3. Why is sin(n/6) aperiodic?
   A. ω0/2π = 1/(12π) is irrational
   B. The amplitude is 1
   C. It is a sine
   D. n/6 is a fraction
   Answer: A. No integer N makes N/6 a multiple of 2π. (Key point: Periodicity of sin(n/6))

4. How is sin(n/6) classified?
   A. Energy signal
   B. Power signal with P = 0.5
   C. Neither
   D. Power signal with P = 1
   Answer: B. E = ∞ because samples never die out; the average of sin^2 is 1/2. (Key point: Classification of sin(n/6))

## Close variants set in other exams

- [Energy and power: 4sin(2πt), cos(3n/4)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-4sin-cos) — Signals and Systems, chapter 1, asked 2×

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