For what values of x does the graph of f(x) = x − 2 sin x have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)

Question

zepdrix · Answer

Hey Sid!
Welcome to OpenStudy!

Recall that your derivative function will give you the slope of the line tangent to the curve at a given point.

So we're concerned with zero slope, horizontal tangent lines.
This happens when $\large\rm f'(x)=0$.

Do you understand how to find your derivative function from what was given?

itz_sid · Answer

Yea so, f'(x) = 1-2Cosx.

itz_sid · Answer

And then I have to solve for x?

zepdrix · Answer

Ok great.
Then set your derivative function equal to 0,
this is what it means to look for horizontal tangents,$$\large\rm 0=1-2\cos x$$And then yes, solve for x.
Remember some of your trig stuff? :)

itz_sid · Answer

Yea, haha.
so $$\frac{ \pi }{ 2 }$$ ?

itz_sid · Answer

oh wait sorry

itz_sid · Answer

$$\frac{ \pi }{ 3 }$$

zepdrix · Answer

So umm you ended up with 1/2 on the other side, right?
1/2 is the shorter length... and cosine is the x coordinate... sooo
ya ya ya that happens up at pi/3.

But within one rotation of the unit circle, there is one other place.

itz_sid · Answer

Oh yea, and $$\frac{ 5\pi }{ 3 }$$

zepdrix · Answer

Ok, good.
We have something else to worry about as well.

Our function is not restricted to a single rotation of the unit circle [0,2pi].
We can spin around the circle any number of times and get this same 1/2 value.

Example, from pi/3 if we spin 2pi, we land at the same location, giving us the same 1/2 from our cosine, ya?$$\large\rm \cos\left(\frac{\pi}{3}\right)=\cos\left(\frac{\pi}{3}+2\pi\right)=\frac12$$

itz_sid · Answer

Oh yea. :/

zepdrix · Answer

So we have to allow for full rotations.
Any number of full rotations can be added to our solutions.$$\large\rm x=\frac{\pi}{3}+2n \pi$$Where n is an integer value (as the instructions told us to use).

That would be part of our solution, ya?

itz_sid · Answer

Oh i get it. So would that be the answer?

zepdrix · Answer

That would be your first "set of solutions",
we have to include the other set as well.
The 5pi/3 with rotations allowed.

itz_sid · Answer

Oh I see

zepdrix · Answer

Just so it's clear, this is one set of solutions, $\large\rm \dfrac{\pi}{3}+2n\pi$.
You would put a comma before writing the other set of solutions.