Help with derivatives of sin

Find an equation of a line that is tangent to y=2sinx and whose slope is a maximum value.

Thanks for your help! I greatly appreciate it !

y = 2 sin(x)

The slope of the tangent line is
y' = 2 cos(x)

If you want the slope to be maximized (optimized) then
y'' = 0 = -2 sin(x)
0 = sin(x)

x = arcsin(0) = πn , where n∈N

But these include both maximum and minimum optimizations
We narrow it down to x=2πn for maximums by ensuring y''' = -2 cos(x) remain negative

This means the slope will be y'(2πn) = 2 cos(2πn) = 2

And this occurs at y-values of y(2πn) = 2 sin(2πn) = 0

The equation of the line is therefore:
y − y₁ = m(x − x₁)
y − 0 = 2(x − 2πn)
y = 2x − 4πn