Question

For what values of x in [0, 2*pi] does the graph of y={cos(x)}/{2+sin(x)} have a horizontal tangent? List the smaller value of x first.

Answer 1

Horizontal tangent implies a line parallel to the x-axis which has a slope of zero.
The derivative gives the slope.
Find the derivative, equate it to zero and solve for x in the interval [0, 2pi].

Answer 2

okay so the derivative is y'(x) = -(2sinx+1)/(sinx+2)^2

Answer 3

looks good
i bet you used the pythagorean identity cos^2(x)+sin^2(x)=1 to clean it up into that.

Answer 4

so as @aum was saying you need find when y'=0

Answer 5

so when does 2sin(x)+1=0

Answer 6

You will find the unit circle most helpful here.

Answer 7

so if I isolate the x it would be sin^-1(1/2) = x.
What am I looking for in the unit circle?

Answer 8

So is it sqrt(3)/2 and the negative of that?

Answer 9

2sin(x)+1=0
sin(x)=-1/2
There should be two values such that sin(x) is -1/2

Answer 10

sin(x) = -1/2.
From the unit circle, the reference angle is pi/6 radians as sin(pi/6) = 1/2
Sine is negative in the third and fourth quadrant.
Therefore the angles are pi + pi/6 = 7pi/6 and 2pi - pi/6 = 11pi/6

Answer 11

Thanks.