At what point(s)in the interval $[ 0,2 \pi ]$ , does the curve given by $y = 7 \ln ( \sin x )$ have horizontal tangents? Round to two decimal places.
A)0.83, 0.17
B)1.57, 4.71
C)0, 3.14
D)There are no horizontal tangents.

Question

Quizplus · Accepted Answer

To find the points where the curve $y = 7 \ln(\sin x)$ has horizontal tangents, we need to find where its derivative equals zero. The derivative of $y$ with respect to $x$ is:$$y' = 7 \cdot \frac{1}{\sin x} \cdot \cos x = 7 \cot x$$Setting $y'$ equal to zero to find horizontal tangents:$$7 \cot x = 0$$This equation is satisfied when $\cot x$ is undefined, which occurs when $\sin x = 0$. However, within the given interval $[0, 2\pi]$, $\sin x = 0$ at $x = 0, \pi, 2\pi$, but these are not the points we're looking for since $\ln(\sin x)$ would be undefined there. Instead, we look for where $\cos x = 0$, because $\cot x = \frac{\cos x}{\sin x}$, and $\cot x = 0$ when $\cos x = 0$, which occurs at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ within the given interval. These points correspond to $x = 1.57$ and $x = 4.71$ when rounded to two decimal places.

At What Point(s)in the Interval [0,2π][ 0,2 \pi ][0,2π] , Does the Curve Given By

At What Point(s)in the Interval $[ 0,2 \pi ]$ , Does the Curve Given By