Question

hey guys i need to find all values of x for which the tangent of f(x)=x-sinx is horizontal

Answer 1

would the derivative be 1-2cos(2x)?

Answer 2

how did you see which to double angle?

Answer 3

switch*

Answer 4

oh i made a typo in the question it should be f(x)=x-sin2x

Answer 5

So yeah you did good on the derivative part
\[1-2\cos(2x)=0 \\ -2\cos(2x)=-1 \\ \cos(2x)=\frac{1}{2}\]

Answer 6

now find when cos(u)=1/2 on the unit circle

Answer 7

pi/3 and 2pi/3

Answer 8

should i multiply them by 2? because of the 2x?

Answer 9

pi/3 and 5pi/3

Answer 10

true 5pi/3

Answer 11

\[\text{ so we had } \cos(u)=\frac{1}{2} \text{ and found } \\ u=\frac{\pi}{3}+2\pi n \text{ or } u=\frac{5 \pi}{3}+2n \pi \text{ but } u =2x\]

Answer 12

\[2x=\frac{\pi}{3}+2\pi n \text{ or } 2x=\frac{5 \pi}{3}+2 n \pi \]

Answer 13

to find x you divide both sides by 2

Answer 14

so it would be pi/6+2pi(n) or 5pi/6+2pi(n)