Find all x values on the graph of f(x) = x − 2 cos(x) for 0 < x < 2π where the tangent line has slope 2. (Enter your answers as a comma-separated list.)

Answer 1

f(x) = x - 2cosx

Slope of tangent at (x,f(x)) is f'(x) = 1 + 2sinx

Slope = 2 when 1 + 2sinx = 2.

So, sinx = 1/2

x = π/6 or 5π/6

Answer 2

y' = 1 + 2 sin x

2 = 1 + 2 sin x

1 = 2 sinx

1/2 = sin x

x = 30 = pi/6 , x = 150 = 5*pi/6

Answer 3

The slope of the tangent line is, of course, the derivative of f(x). So f '(x) = 1 + 2sin(x) and solving 1 + 2sin(x) = 2 we get:

1 + 2sin(x) = 2

2sin(x) = 1

sin(x) = 1/2

Using the unit circle, the sin(x) is 1/2 when x = π/6 and x = 5π/6