Question

If \( x = r \cos \theta, y = r \sin \theta \), then \( \frac{dy}{dx} \) at \( \theta = \frac{\pi}{4} \), where \( r \) is a constant and \( \theta \) is a parameter, is equal to:

• A. 0
• B. 1
• C. -1
• D. \( \frac{\sqrt{2}}{2} \)
• E. \( \frac{1}{\sqrt{2}} \)

Hint:

The derivative of \( y = r \sin \theta \) and \( x = r \cos \theta \) gives the rate of change of \( y \) with respect to \( x \).

Answer 1

The Correct Option is C

We are given that \( x = r \cos \theta \) and \( y = r \sin \theta \). To find \( \frac{dy}{dx} \), we differentiate \( x \) and \( y \) with respect to \( \theta \): \[ \frac{dx}{d\theta} = -r \sin \theta, \quad \frac{dy}{d\theta} = r \cos \theta. \] Thus, the derivative \( \frac{dy}{dx} \) is given by: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{r \cos \theta}{-r \sin \theta} = -\cot \theta. \] At \( \theta = \frac{\pi}{4} \), \( \cot \left( \frac{\pi}{4} \right) = 1 \), so: \[ \frac{dy}{dx} = -1. \] Thus, the value of \( \frac{dy}{dx} \) at \( \theta = \frac{\pi}{4} \) is -1.