---
title: "Pedal equation of r^m = a^m cos mθ"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:pedal-equation-rm-cos-m-theta"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/pedal-equation-rm-cos-m-theta
site: BCT Engineering
---

# Pedal equation of r^m = a^m cos mθ

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Asoj; 2068 Chaitra (asked 2 times)
- Video: [Find the Pedal Equation for the given Polar Curve r^m=a^m(cosm(theta)+sinm(theta))](https://www.youtube.com/watch?v=Y75kSfJPUWg) — Learn With Sumi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/pedal-equation-rm-cos-m-theta
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the pedal equation of the polar curve r^m = a^m cos mθ.

## Short answer

The pedal equation is a^m p = r^(m+1). Using φ = π/2 + mθ gives p = r cos mθ, then substitute cos mθ = r^m / a^m.

## Answer

The pedal equation is a relation between r and p, where p is the perpendicular from the pole to the tangent. Use p = r sin φ, where tan φ = r dθ/dr.

Take logs of r^m = a^m cos mθ:

m ln r = m ln a + ln cos mθ

Differentiate with respect to θ:

(m/r)(dr/dθ) = −m tan mθ, so (1/r)(dr/dθ) = −tan mθ

```text
tan φ = r dθ/dr = −cot mθ = tan(π/2 + mθ)
so   φ = π/2 + mθ
```
Then

p = r sin φ = r sin(π/2 + mθ) = r cos mθ

From the curve, cos mθ = r^m / a^m. Hence

p = r^(m+1) / a^m

**Pedal equation: a^m p = r^(m+1).**

## Check yourself

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

1. A pedal equation relates:
   A. r and p, p being the perpendicular from pole to tangent
   B. x and y
   C. p and theta
   D. r and theta
   Answer: A. It connects radius vector and perpendicular distance. (Key point: Pedal equation meaning)

2. For r^m = a^m cos m theta, phi equals:
   A. pi + m theta
   B. pi/2 + m theta
   C. m theta
   D. pi/2 - m theta
   Answer: B. tan phi = -cot(m theta) = tan(pi/2 + m theta). (Key point: Angle phi)

3. p = r sin(pi/2 + m theta) =
   A. r cos m theta
   B. r sin m theta
   C. r tan m theta
   D. r
   Answer: A. sin(pi/2 + x) = cos x. (Key point: Value of p)

4. Using cos m theta = r^m/a^m, the pedal equation is:
   A. a^m p = r^(m+1)
   B. a^m p = r^m
   C. p = r^(m-1)
   D. p = a^m r^(m+1)
   Answer: A. p = r * r^m/a^m. (Key point: Result)

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