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

# z-transform of aᵏ cos kπ

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2069 Poush
- Video: [Part-1 Z transform of trigonometric function [cos(w0n).u(n)]](https://www.youtube.com/watch?v=Vw9i4jvy8v0) — VKY Academy
- 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-a-k-cos-k-pi
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of aᵏ cos kπ for k ≥ 0.

## Short answer

cos(kπ)=(-1)^k, so a^k cos kπ = (-a)^k. Using Z[b^k]=z/(z-b) gives Z[a^k cos kπ] = z/(z+a), |z|>|a|.

## Answer

**Simplify first.** cos kπ = (−1)ᵏ, so

aᵏ cos kπ = (−a)ᵏ.

Use Z[bᵏ] = z/(z − b) with b = −a:

**Answer:** Z[aᵏ cos kπ] = z/(z + a), for |z| > |a|.

**Cross-check with the general formula.** Z[aᵏ cos kθ] = z(z − a cos θ)/(z² − 2az cos θ + a²). Put θ = π (cos π = −1, sin π = 0):

```text
z (z + a) / (z² + 2az + a²) = z (z + a) / (z + a)² = z/(z + a)  ✓
```

## Check yourself

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

1. cos(k pi) equals
   A. (-1)^k
   B. 1
   C. 0
   D. (-1)^(k+1)
   Answer: A. cos pi = -1, cos 2pi = 1, alternating. (Key point: Simplification)

2. a^k cos(k pi) is the same as
   A. a^k
   B. (-a)^k
   C. a^(2k)
   D. (1/a)^k
   Answer: B. a^k (-1)^k = (-a)^k. (Key point: Combined sequence)

3. Z[a^k cos k pi] =
   A. z/(z - a)
   B. z(z - a)/(z + a)^2
   C. z/(z + a), |z| > |a|
   D. a z/(z + a)
   Answer: C. Z[b^k] = z/(z - b) with b = -a. (Key point: Result)

4. Putting theta = pi in z(z - a cos theta)/(z^2 - 2az cos theta + a^2) gives
   A. z(z - a)/(z - a)^2
   B. z/(z - a)
   C. z(z + a)/(z - a)^2
   D. z(z + a)/(z + a)^2 = z/(z + a)
   Answer: D. cos pi = -1 turns denominator into (z+a)^2. (Key point: Cross-check)

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