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

# z-transform of cos at

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2074 Bhadra, 2 marks; 2073 Bhadra, 1.5 marks; 2070 Magh (asked 3 times)
- Video: [Z-Transform of COS Signal Type-1](https://www.youtube.com/watch?v=zczEFJD4MXw) — TutorialsPoint
- 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-cos-at
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of cos at.

## Short answer

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

## Answer

Take sampling period T, so x(kT) = cos(akT). Use Euler's formula:

cos at = (e^(iat) + e^(−iat))/2

and Z[e^(±iat)] = z/(z − e^(±iaT)) (ROC |z| > 1).

Z[cos at] = ½ [ z/(z − e^(iaT)) + z/(z − e^(−iaT)) ]
= ½ · z [ (z − e^(−iaT)) + (z − e^(iaT)) ] / [ (z − e^(iaT))(z − e^(−iaT)) ]

Numerator: z[2z − 2cos aT]. Denominator: z² − z(e^(iaT) + e^(−iaT)) + 1 = z² − 2z cos aT + 1.

**Answer:**

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

(For T = 1: z(z − cos 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 cos at transformed?
   A. Using Z[t x(t)]
   B. Euler: cos at = (e^{iat}+e^{-iat})/2, using Z[e^{±iat}] = z/(z-e^{±iaT})
   C. Using final value theorem
   D. Using partial fractions of z²+1
   Answer: B. Linearity. (Key point: Technique)

2. What is the denominator after combining?
   A. z² + 2z cos aT + 1
   B. z² - 2z sin aT + 1
   C. z² - 2z cos aT + 1
   D. z² - 1
   Answer: C. Product (z-e^{iaT})(z-e^{-iaT}). (Key point: Denominator)

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

## Related in Applied Mathematics

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## Related in other subjects

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