Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = \cos \theta + \sin 2 \theta + 8 , \quad y = \sin \theta + \cos 2 \theta + 8 , \quad \theta = \pi$
A) $y = \frac { x } { 2 } + 2$
B) $y = \frac { 2 } { x }$
C) $y = \frac { x } { 2 } + \frac { 3 } { 2 }$
D) $y = \frac { 25 } { 2 } - \frac { x } { 2 }$
E) None of these

Question

Quizplus · Accepted Answer

At $\theta = \pi$, $x = 8$ and $y = 7$. The derivatives $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$ are $-\sin \theta + 2\cos 2\theta$ and $\cos \theta - 2\sin 2\theta$ respectively. At $\theta = \pi$, $\frac{dx}{d\theta} = -1$ and $\frac{dy}{d\theta} = -1$, so the slope of the tangent is 1. Using point-slope form, the equation is $y - 7 = 1(x - 8)$, simplifying to $y = x - 1$, which matches option D when rearranged.

Find an Equation of the Tangent to the Curve at the Point