Question

The tangent to the curve y=x+sin x has a horizontal tangent when x =

Answer 1

A 'tangent' to a line means the derivative. You can differentiate the function as follows:
$f(x) = x+sin(x)$
$f'(x) = 1+cos(x)$

At a horizontal tangent, the derivative is equal to 0. So, let's set it equal to 0, and solve for x. Remember, whenever doing trigonometry in calculus, we assume we are answering in radians. So:
$f'(x) = 1+cos(x)$
$0 = 1+cos(x)$
$-1=cos(x)$

The values at which cos(x) equals -1 are π, and then 3π, 5π and so on. You basically just keep adding 2π to the previous number and get the answer. So, this can be written as:
$a_n = a_{(n-1)}+2 \pi$

That will be your final answer. You could write this as an infinite series, but I just didn't want to complicate it. So, if you want one, I can give you one.