---
title: "Fourier cosine transform"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:fourier-cosine-transform"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-cosine-transform
site: BCT Engineering
---

# Fourier cosine transform

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Kartik (new course), 2 marks
- Video: [Fourier Sine and Cosine Transform (Concept Overview and solved examples)](https://www.youtube.com/watch?v=enUsWxdiczA) — Mathematics for Engineers 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-cosine-transform
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Determine the Fourier cosine transform of f(x) = 1 for 0 < x < a; 0 otherwise.

## Short answer

F_c(ω) = ∫_0^a cos ωx dx = sin(aω)/ω.

## Answer

The Fourier cosine transform of f(x) on x > 0 is F_c(ω) = ∫₀^∞ f(x) cos ωx dx.

For f(x) = 1 on 0 < x < a and 0 for x > a:

```text
F_c(ω) = ∫₀^a cos ωx dx = [ sin ωx / ω ]₀^a
```

**Answer:** F_c(ω) = sin(aω) / ω

In the symmetric (normalised) convention, F_c(ω) = √(2/π) · sin(aω)/ω. The inverse is f(x) = (2/π)∫₀^∞ [sin(aω)/ω] cos ωx dω.

## Check yourself

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

1. F_c(ω) =
   A. ∫ f sin ωx dx
   B. ∫0^∞ f(x) cos ωx dx
   C. ∫ f e^(iωx) dx
   D. (2/π)∫ f dx
   Answer: B. Standard. (Key point: Definition)

2. For f = 1 on (0,a) the integral runs over:
   A. 0 to ∞
   B. −a to a
   C. 0 to a
   D. 0 to 1
   Answer: C. f is 0 beyond a. (Key point: Limits)

3. F_c(ω) =
   A. cos(aω)/ω
   B. a sin ω
   C. (1 − cos aω)/ω
   D. sin(aω)/ω
   Answer: D. ∫0^a cos ωx dx. (Key point: Result)

## Related in Mathematics III

- [Fourier cosine integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-integral) — Mathematics III, chapter 4
- [Fourier cosine integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-integral-two) — Mathematics III, chapter 4
- [Fourier sine transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-sine-transform) — Mathematics III, chapter 4
- [Fourier transform of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-transform-of-a-function) — Mathematics III, chapter 4
- [Fourier transform of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-transform-of-a-function-two) — Mathematics III, chapter 4
- [Half range cosine series: πx − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-cosine-series-x-x2) — Mathematics III, chapter 4, asked 3×

## Related in other subjects

- [Cosine transform of e^{-x}; Parseval identity](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-parseval-e-x) — Applied Mathematics, chapter 4, asked 3×
- [Cosine transform of e^{-mx}; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-transform-e-mx-beta) — Applied Mathematics, chapter 4, asked 3×
- [Fourier cosine integral of e^{-kx}](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-integral-e-kx) — Applied Mathematics, chapter 4, asked 2×
- [Fourier transform of 1-x²; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-1-x2-evaluate) — Applied Mathematics, chapter 4, asked 2×