---
title: "z-transform of cosh(at) sin(bt)"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-cosh-at-sin-bt"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-cosh-at-sin-bt
site: BCT Engineering
---

# z-transform of cosh(at) sin(bt)

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2069 Bhadra; 2068 Magh, 2.5 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-cosh-at-sin-bt
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of cosh(at) sin(bt).

## Short answer

Using Z[e^{-αt} sin bt] and linearity, Z[cosh at sin bt] = z sin bT [cosh aT (z^2+1) - 2z cos bT] / [z^4 - 4 cos bT cosh aT z^3 + (2 cosh 2aT + 4 cos^2 bT) z^2 - 4 cos bT cosh aT z + 1], ROC |z| > e^{aT}.

## Answer

Take sampling period T. Write cosh at · sin bt = ½ (e^(at) + e^(−at)) sin bt.

Use the standard result Z[e^(−αt) sin bt] = z e^(−αT) sin bT / (z² − 2z e^(−αT) cos bT + e^(−2αT)).

Put α = −a and α = +a:

Z[cosh at sin bt] = ½ { z e^(aT) sin bT / (z² − 2z e^(aT) cos bT + e^(2aT)) + z e^(−aT) sin bT / (z² − 2z e^(−aT) cos bT + e^(−2aT)) }

**Combined form.** Let p = e^(aT), so 1/p = e^(−aT), c = cos bT, s = sin bT. Adding over a common denominator (using p · 1/p = 1):

- Numerator: p(z² − 2zc/p + 1/p²) + (1/p)(z² − 2zcp + p²) = (p + 1/p)(z² + 1) − 4cz = 2 cosh aT (z² + 1) − 4cz
- Denominator: z⁴ − 4c cosh aT · z³ + (2 cosh 2aT + 4c²) z² − 4c cosh aT · z + 1

**Answer:**

Z[cosh at sin bt] = z sin bT [ cosh aT (z² + 1) − 2z cos bT ] / [ z⁴ − 4 cos bT cosh aT z³ + (2 cosh 2aT + 4 cos² bT) z² − 4 cos bT cosh aT z + 1 ]

valid for |z| > e^(aT).

## Check yourself

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

1. How is cosh at sin bt written?
   A. e^{at} sin bt
   B. cosh(a) sin(bt)
   C. ½(e^{at} + e^{-at}) sin bt
   D. ½(e^{at} − e^{-at}) sin bt
   Answer: C. Definition of cosh. (Key point: Decomposition)

2. Which result is used?
   A. Z[sin bt] only
   B. Z[e^{-αt} cos bt]
   C. Z[t sin bt]
   D. Z[e^{-αt} sin bt] = z e^{-αT} sin bT/(z²−2z e^{-αT} cos bT + e^{-2αT}) with α = ∓a
   Answer: D. Apply twice. (Key point: Standard result)

3. What is the ROC?
   A. |z| > e^{aT}
   B. |z| > 1
   C. |z| < e^{aT}
   D. |z| > e^{-aT}
   Answer: A. The e^{at} term grows. (Key point: ROC)

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