---
title: "z-transform of cos kθ and aᵏ cos kθ"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-cos-k-theta-a-k"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-cos-k-theta-a-k
site: BCT Engineering
---

# z-transform of cos kθ and aᵏ cos kθ

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2068 Bhadra
- Video: [Z Transforms of Cos n𝜽 & Sin n𝜽  || 18mat31 || Dr Prashant Patil](https://www.youtube.com/watch?v=ZcoftgMVwh8) — Dr Prashant Patil
- 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-k-theta-a-k
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Show that the Z-transform of cos kθ is z(z - cosθ)/(z² - 2z cosθ + 1). Use this result to find the Z-transform of aᵏ cos kθ.

## Short answer

Use scaling property Z[a^k x(k)]=X(z/a) on Z[cos kθ]=z(z-cosθ)/(z^2-2z cosθ+1) to obtain Z[a^k cos kθ]=z(z - a cosθ)/(z^2 - 2a z cosθ + a^2), |z|>|a|.

## Answer

#### Z-transform of cos kθ

Use Euler's formula cos kθ = (e^{jkθ} + e^{−jkθ})/2 and Z[bᵏ] = z/(z − b), |z| > |b|.

```text
Z[cos kθ] = ½ [ z/(z − e^{jθ}) + z/(z − e^{−jθ}) ]
          = ½ · z [ (z − e^{−jθ}) + (z − e^{jθ}) ]
                / [ (z − e^{jθ})(z − e^{−jθ}) ]
          = ½ · z [ 2z − 2 cos θ ] / [ z² − 2z cos θ + 1 ]
```

**Z[cos kθ] = z (z − cos θ) / (z² − 2z cos θ + 1)**, |z| > 1. ✓

#### Z-transform of aᵏ cos kθ

Scaling (damping) property: if Z[x(k)] = X(z), then Z[aᵏ x(k)] = X(z/a).

Replace z by z/a:

```text
Z[aᵏ cos kθ] = (z/a)(z/a − cos θ) / ( z²/a² − 2(z/a) cos θ + 1 )
             = (z/a)(z − a cos θ)/a  ·  a² / (z² − 2az cos θ + a²)
```

**Answer:** Z[aᵏ cos kθ] = z (z − a cos θ) / (z² − 2a z cos θ + a²), |z| > |a|.

## Check yourself

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

1. Z[cos k theta] starts from
   A. cos k theta = (e^(jk theta) + e^(-jk theta))/2
   B. (e^(jk theta) - e^(-jk theta))/2
   C. (e^(jk theta) + e^(-jk theta))/(2j)
   D. e^(jk theta)
   Answer: A. Sum, divided by 2. (Key point: Euler form)

2. Z[cos k theta] =
   A. z sin theta/(z^2 - 2z cos theta + 1)
   B. z(z - cos theta)/(z^2 - 2z cos theta + 1), |z| > 1
   C. z(z + cos theta)/(z^2 - 2z cos theta + 1)
   D. z(z - cos theta)/(z^2 + 1)
   Answer: B. Numerator 2z - 2cos theta halved. (Key point: Result)

3. How is a^k cos(k theta) obtained?
   A. Replace z by az
   B. Multiply X(z) by a
   C. Replace z by z/a in X(z)
   D. Replace z by z - a
   Answer: C. Z[a^k x(k)] = X(z/a). (Key point: Scaling property)

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