The points $A$ and $B$ lie on the curve $y=\sin (x).$

1. Given that the $x$ coordinate of $A$ is $\frac{2\pi }{3},$ find the equation of the tangent to the curve at the point $A.$

2. The $x$ coordinate of $B$ is $b,$ and $\pi <b<2\pi .$ Given that the normal to the curve at $B$ is perpendicular to the tangent at $A,$ find the coordinates of $B.$

3. The tangent at $A$ and the normal at $B$ intersect at $C.$

   Find, in exact form, the coordinates of $C.$

Here is a picture:

To find the coordinates of $B$, notice that if the normal at $B$ is perpendicular to the tangent at $A$, then the tangent at $B$ must be paralle to the tangent at $A$.

In other words, $\frac{dy}{dx}$ at $A$ is the same as $\frac{dy}{dx}$ at $B$.

The normal at $B$ is perpendicular to the tangent at $B,$ so use $-\frac{1}{m}.$

If the coordinates of $C$ are $(x,y)$, then $x+y$ can be expressed in simplest form as

Give the value of $ab$