---
title: "Energy of 5cosπt + sin5πt"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-of-sum-of-sinusoids"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-of-sum-of-sinusoids
site: BCT Engineering
---

# Energy of 5cosπt + sin5πt

Signals and Systems · Chapter 1: Signals

- Asked in: 2078 Chaitra, 4 marks
- 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-of-sum-of-sinusoids
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Determine the energy of the signal x(t) = 5cosπt + sin5πt, −∞ < t < ∞

## Short answer

x(t)=5cosπt+sin5πt is periodic with period 2 s. Using orthogonality over one period, the energy per period is 26. Over infinite time, energy is infinite: E = ∞. (It is a power signal with P = 13 W.)

## Answer

$x(t) = 5\cos\pi t + \sin 5\pi t$ exists for all time and does not decay, so its energy is expected to be infinite. Check with the definition.

Periods: $\cos\pi t$ has $T_1 = 2$ s, $\sin 5\pi t$ has $T_2 = 2/5$ s; the ratio is rational, so $x(t)$ is periodic with $T_0 = \text{LCM}(2, 0.4) = 2$ s.

$$
x^2(t) = 25\cos^2\pi t + \sin^2 5\pi t + 10\cos\pi t\,\sin 5\pi t
$$

Energy in one period:

$$
\begin{aligned}
E_{T_0} &= \int_0^{2}25\cos^2\pi t\,dt + \int_0^{2}\sin^2 5\pi t\,dt + 10\int_0^{2}\cos\pi t\sin5\pi t\,dt \\
&= 25(1) + 1 + 0 = 26
\end{aligned}
$$

(The cross term is zero because sinusoids of different frequencies are orthogonal over a common period.)

Total energy over $-\infty < t < \infty$ = (number of periods) × 26:

$$
E = \lim_{k\to\infty} k \times 26 = \infty
$$

Average power:

$$
P = \frac{E_{T_0}}{T_0} = \frac{26}{2} = 13 \quad\left(= \frac{5^2}{2} + \frac{1^2}{2}\right)
$$

**Answer:** $E = \infty$; the signal is a **power signal** with $P = 13$ W (normalised).

## Check yourself

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

1. The period of 5 cos(pi t) + sin(5 pi t) is:
   A. 2 s
   B. 0.4 s
   C. 1 s
   D. 10 s
   Answer: A. Periods 2 s and 0.4 s have LCM 2 s. (Key point: Period)

2. The cross term 10 cos(pi t) sin(5 pi t) integrates over a period to:
   A. 10
   B. 0
   C. 5
   D. 25
   Answer: B. Sinusoids of different frequencies are orthogonal over a common period. (Key point: Cross term)

3. Energy in one period is 26, so average power is:
   A. 26
   B. 13
   C. 52
   D. 1/13
   Answer: B. P = 26/2 = 13, equal to 25/2 + 1/2. (Key point: Energy and power)

4. Total energy over all time is:
   A. 26
   B. 13
   C. Infinite
   D. 0
   Answer: C. Infinitely many periods each add 26. (Key point: Total energy)

## Related in Signals and Systems

- [DT periodicity; energy/power of 5cosπt + sin5πt](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-periodicity-energy-power-sinusoids) — Signals and Systems, chapter 1
- [Sketch 5cos(t); energy or power](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asketch-cosine-energy-power) — Signals and Systems, chapter 1
- [Power of energy signal is zero](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-signal-example) — Signals and Systems, chapter 1

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