Question

The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.

• A. \( -2\sqrt{3} \)
• B. \( \frac{-2}{\sqrt{3}} \)
• C. \( -2 \)
• D. \( -\frac{1}{2} \)

Hint:

For parametric curves \( x = f(\theta) \), \( y = g(\theta) \), the slope is: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]

Answer 1

The Correct Option is C

Step 1: Find \( \frac{dy}{dx} \)
\[ \frac{dx}{d\theta} = \cos\theta, \quad \frac{dy}{d\theta} = -2\sin 2\theta \] \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{-2\sin 2\theta}{\cos\theta} \] Step 2: Evaluate at \( \theta = \frac{\pi}{6} \)
\[ \sin 2\theta = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}, \quad \cos\theta = \frac{\sqrt{3}}{2} \] \[ \frac{dy}{dx} = \frac{-2 \times \frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}} = -2 \]