Find the x-coordinates of all points on the curve y=2sin-sin^2x at which the tangent line is horizontal.
To find the x-coordinates of all points on the curve where the tangent line is horizontal, we need to find the derivative of the function y = 2sin(x) - sin^2(x) and set it equal to zero.
Let's start by finding the derivative of y with respect to x:
dy/dx = d/dx [2sin(x) - sin^2(x)]
= 2cos(x) - 2sin(x)cos(x)
Next, set the derivative equal to zero and solve for x:
2cos(x) - 2sin(x)cos(x) = 0
Factor out 2cos(x):
2cos(x)[1 - sin(x)] = 0
Now we have two possibilities:
1. 2cos(x) = 0
This equation holds true when cos(x) = 0.
The x-values where cos(x) = 0 are π/2 + nπ, where n is an integer.
2. 1 - sin(x) = 0
This equation holds true when sin(x) = 1.
The x-values where sin(x) = 1 are π/2 + 2nπ, where n is an integer.
Combining both possibilities, the x-coordinates of all points on the curve where the tangent line is horizontal are:
π/2 + nπ, π/2 + 2nπ, where n is an integer.