---
title: "Laplace transforms of functions cos³2t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-cos32t"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-cos32t
site: BCT Engineering
---

# Laplace transforms of functions cos³2t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2076 Chaitra, 1+1.5+2.5 marks
- Video: [10. L[cos^3 (2t)]], L[sinh^3 (4t)]],](https://www.youtube.com/watch?v=xZJL3N5yxHE) — Engineering Vidya
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-cos32t
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

State the condition for existence of the Laplace transform. Obtain the Laplace transform of: a) cos³ 2t  b) (cos at − cos bt)/t.

## Short answer

Existence condition: f(t) piecewise continuous and of exponential order, |f(t)| ≤ M e^{kt}. a) L{cos^3 2t} = (1/4)[3s/(s^2+4) + s/(s^2+36)]. b) L{(cos at - cos bt)/t} = (1/2) ln((s^2+b^2)/(s^2+a^2).

## Answer

**Existence condition.** The Laplace transform L{f(t)} = ∫₀^∞ e^(−st) f(t) dt exists for s > k if f(t) is piecewise continuous on every finite interval [0, T] and of exponential order, that is |f(t)| ≤ M e^(kt) for all t ≥ T. (These conditions are sufficient, not necessary.)

#### a) L{cos³ 2t}
Use cos³x = (3 cos x + cos 3x)/4 with x = 2t: cos³ 2t = (3 cos 2t + cos 6t)/4.

```text
L{cos³ 2t} = ¼ [3s/(s² + 4) + s/(s² + 36)]
```

#### b) L{(cos at − cos bt)/t}
Use L{f(t)/t} = ∫ₛ^∞ F(u) du with F(s) = s/(s² + a²) − s/(s² + b²).

```text
∫ₛ^∞ [u/(u² + a²) − u/(u² + b²)] du = ½ [ln((u² + a²)/(u² + b²))]ₛ^∞
 = −½ ln((s² + a²)/(s² + b²))
 = ½ ln((s² + b²)/(s² + a²))
```

**Answers:** a) ¼[3s/(s² + 4) + s/(s² + 36)]; b) ½ ln((s² + b²)/(s² + a²)).

## Check yourself

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

1. Which conditions guarantee that L{f(t)} exists for s > k?
   A. f is differentiable everywhere and bounded
   B. f is periodic and continuous at t = 0
   C. f is continuous with f(0) = 0 and f(t) -> 0 as t -> infinity
   D. f is piecewise continuous on every finite interval and of exponential order |f(t)| <= M e^(kt)
   Answer: D. The answer states these two conditions: piecewise continuity on each [0, T] and exponential order. They are sufficient, and the transform then exists for s > k. (Key point: Conditions for the Laplace transform to exist)

2. cos^3 x equals what?
   A. (3 cos x + cos 3x)/4
   B. (3 cos x - cos 3x)/4
   C. (cos x + 3 cos 3x)/4
   D. (1 + cos 2x)/2
   Answer: A. This is the standard reduction of cos^3. (Key point: Triple-angle identity)

3. What is L{cos^3 2t}?
   A. (1/4)[3s/(s^2+4) + s/(s^2+9)]
   B. (1/4)[3s/(s^2+4) + s/(s^2+36)]
   C. (1/4)[3s/(s^2+4) + 3s/(s^2+36)]
   D. (1/4)[3s/(s^2+4) - s/(s^2+36)]
   Answer: B. cos^3 2t = (3 cos 2t + cos 6t)/4, so frequencies 2 and 6. (Key point: Result for (a))

4. What is L{(cos at - cos bt)/t}?
   A. (1/2) ln((s^2+a^2)/(s^2+b^2))
   B. ln((s^2+b^2)/(s^2+a^2))
   C. (1/2) ln((s^2+b^2)/(s^2+a^2))
   D. (1/2)(s/(s^2+a^2) - s/(s^2+b^2))
   Answer: C. Integrating from s to infinity flips the ratio and gives a factor 1/2. (Key point: Result for (b))

## Close variants set in other exams

- [Laplace transforms of functions 1/√t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-1-t) — Mathematics III, chapter 3, asked 4×
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