---
title: "Sketch 5cos(t); energy or power"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:sketch-cosine-energy-power"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/sketch-cosine-energy-power
site: BCT Engineering
---

# Sketch 5cos(t); energy or power

Signals and Systems · Chapter 1: Signals

- Asked in: 2080 Chaitra, 4 marks
- Video: [| Check whether the following signals are energy or power signals |](https://www.youtube.com/watch?v=p-VnV_FnxF8) — Dr.Santosh Tondare Engineering Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/sketch-cosine-energy-power
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Sketch the signal x(t) = 5cos(t). State whether the given signal is energy signal or power signal.

## Short answer

The signal is a cosine of amplitude 5, period 2π s, oscillating between ±5. It is a power signal with average power P = 25/2 = 12.5 W (normalized) and infinite energy.

## Answer

$x(t) = 5\cos t$ has amplitude 5 and $\omega_0 = 1$ rad/s, so period $T_0 = 2\pi/1 = 2\pi \approx 6.28$ s. Peaks of $+5$ at $t = 0, \pm2\pi, \ldots$; zeros at $t = \pm\pi/2, \pm3\pi/2$; minimum $-5$ at $t = \pm\pi$.

```text
 x(t)
  5 +.                   .
    | '.               .'
  0 +---'.---------.'-------> t
    |  pi/2 '.   .' 3pi/2
 -5 +         '-'
    0         pi        2pi
```

(The waveform repeats every $2\pi$ s for all $t$, also for negative $t$, since cosine is even.)

#### Energy or power

**Energy:**

$$
E = \int_{-\infty}^{\infty}25\cos^2 t\,dt = \infty
$$

**Power** (periodic signal, average over one period):

$$
\begin{aligned}
P &= \frac{1}{2\pi}\int_0^{2\pi}25\cos^2 t\,dt = \frac{25}{2\pi}\int_0^{2\pi}\frac{1+\cos 2t}{2}\,dt \\
&= \frac{25}{2\pi}\cdot\pi = \frac{25}{2} = 12.5
\end{aligned}
$$

**Answer:** $x(t) = 5\cos t$ is a **power signal** with $P = 12.5$ W (normalised to 1 Ω) and $E = \infty$.

## Check yourself

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

1. The period of 5 cos t is:
   A. 2 pi
   B. pi
   C. 5
   D. 10 pi
   Answer: A. w0 = 1 rad/s so T0 = 2 pi. (Key point: Period)

2. 5 cos t has a minimum of -5 at:
   A. t = 0
   B. t = pi/2
   C. t = pi
   D. t = 2 pi
   Answer: C. cos pi = -1, so the minimum is -5 at t = pi. (Key point: Key points of waveform)

3. The average power is:
   A. 25
   B. 12.5
   C. 5
   D. 2.5
   Answer: B. P = A^2/2 = 25/2 = 12.5. (Key point: Power)

4. The signal is a power signal because:
   A. E = infinity and 0 < P < infinity
   B. E is finite
   C. P = 0
   D. It is even
   Answer: A. Periodic everlasting signals have infinite energy and finite power. (Key point: Classification)

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