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

# z-transform of sin kθ and aᵏ sin kθ

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2071 Magh, 5 marks
- Video: [Part-2 Z transform of trigonometric function [sin(w0n).u(n)]](https://www.youtube.com/watch?v=emGFpV3YHsw) — 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-sin-k-theta-a-k
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of sin kθ. Use it to find Z[aᵏ sin kθ].

## Short answer

Z[sin kθ] = z sinθ/(z^2 - 2z cosθ + 1). Scaling property then gives Z[a^k sin kθ] = a z sinθ/(z^2 - 2a z cosθ + a^2), |z|>|a|.

## Answer

#### Z-transform of sin kθ

sin kθ = (e^{jkθ} − e^{−jkθ})/(2j), and Z[bᵏ] = z/(z − b).

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

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

#### Z-transform of aᵏ sin kθ

By the scaling property, Z[aᵏ x(k)] = X(z/a). Replace z by z/a:

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

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

(Numerical check, a = 0.5, θ = 0.7, z = 3: sum of series = 0.138930, formula = 0.138930.)

## Check yourself

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

1. sin k theta is written as
   A. (e^(jk theta) - e^(-jk theta))/(2j)
   B. (e^(jk theta) + e^(-jk theta))/(2j)
   C. (e^(jk theta) - e^(-jk theta))/2
   D. (e^(jk theta) + e^(-jk theta))/2
   Answer: A. Euler's formula for sine. (Key point: Euler form)

2. Z[sin k theta] =
   A. z cos theta/(z^2 - 2z cos theta + 1)
   B. z sin theta/(z^2 - 2z cos theta + 1), |z| > 1
   C. z sin theta/(z^2 - 2z sin theta + 1)
   D. sin theta/(z^2 - 2z cos theta + 1)
   Answer: B. 2j sin theta over 2j gives sin theta. (Key point: First result)

3. Z[a^k x(k)] equals
   A. X(az)
   B. a X(z)
   C. X(z/a)
   D. X(z - a)
   Answer: C. Replace z by z/a. (Key point: Scaling)

4. Z[a^k sin k theta] =
   A. z sin theta/(z^2 - 2az cos theta + a^2)
   B. a z sin theta/(z^2 - 2z cos theta + 1)
   C. a z cos theta/(z^2 - 2az cos theta + a^2)
   D. a z sin theta/(z^2 - 2az cos theta + a^2), |z| > |a|
   Answer: D. (z/a) sin theta over (z^2/a^2 - ...) times a^2. (Key point: Final result)

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