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

# z-transform of sin at

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Magh, 1.5 marks; 2072 Asoj; 2070 Bhadra (asked 3 times)
- Video: [sin(wn).u(n) - z transform and ROC - signals and systems](https://www.youtube.com/watch?v=FcTWNhb7DNw) — faculty 4u
- 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-sin-at
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of sin at.

## Short answer

Z[sin at] = z sin aT / (z^2 - 2z cos aT + 1) for |z| > 1. With T=1, it is z sin a/(z^2 - 2z cos a + 1).

## Answer

Take sampling period T, so x(kT) = sin(akT). Using sin at = (e^(iat) − e^(−iat))/(2i) and Z[e^(±iat)] = z/(z − e^(±iaT)) for |z| > 1:

Z[sin at] = (1/2i) [ z/(z − e^(iaT)) − z/(z − e^(−iaT)) ]
= (1/2i) · z [ (z − e^(−iaT)) − (z − e^(iaT)) ] / [ (z − e^(iaT))(z − e^(−iaT)) ]

Numerator: z (e^(iaT) − e^(−iaT)) = z · 2i sin aT. Denominator: z² − 2z cos aT + 1.

**Answer:**

Z[sin at] = z sin aT / (z² − 2z cos aT + 1),  |z| > 1

(For T = 1: z sin a/(z² − 2z cos a + 1).)

## Check yourself

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

1. How is sin at transformed?
   A. sin at = (e^{iat}-e^{-iat})/(2i), then use Z[e^{±iat}]
   B. (e^{iat}+e^{-iat})/2
   C. Using Z[t x(t)]
   D. Using initial value theorem
   Answer: A. Euler identity. (Key point: Technique)

2. What is the numerator after combining?
   A. z · 2 cos aT
   B. z · 2i sin aT, which cancels with the 2i
   C. z · sin aT without 2i
   D. z² sin aT
   Answer: B. e^{iaT}-e^{-iaT} = 2i sin aT. (Key point: Numerator)

3. What is Z[sin at]?
   A. z cos aT/(z² - 2z cos aT + 1)
   B. z(z - sin aT)/(z² - 2z cos aT + 1)
   C. z sin aT/(z² - 2z cos aT + 1)
   D. sin aT/(z² - 2z cos aT + 1)
   Answer: C. ROC |z|>1. (Key point: Result)

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