Question

cos^3 x = cos x

Find all solutions of the equation in the interval
[0, 2π). (Enter your answers as a comma-separated list.

Answer 1

\[\large\rm \cos^3x=\cos x\]Subtract cosx from each side,\[\large\rm \cos^3x-\cos x=0\]Next, factor a cosx out of each term.

Answer 2

Can you tell me what that would look like?

Answer 3

cos x (cos x +1) (cos x -1)

Answer 4

So you applied your difference of squares after factoring the cosx out?
Ahh good good.

Answer 5

\[\large\rm \cos x(\cos x+1)(\cos x-1)=0\]Next apply your Zero-Factor Property,\[\large\rm \cos x=0\]\[\large\rm (\cos x+1)=0\]\[\large\rm (\cos x-1)=0\]And solve for x in each of these cases.

Answer 6

Is the answer pi and 2pi?

Answer 7

Taking this third part,
\[\large\rm \cos x-1=0\]adding 1,\[\large\rm \cos x=1\]Cosine gives us 1 at an angle of 0.

So x=0 is a solution.
Which means, yes, 2pi would also be a solution (because 0 and 2pi are co-terminal).

But notice that 2pi is not in our interval.
When we have a rounded bracket on the interval, it means we `exclude` that particular end value.

Answer 8

\[\large\rm \cos x+1=0\]subtracting 1,\[\large\rm \cos x=-1\]Cosine is negative 1 when we're on the left side of the x-axis, so yes pi.

Answer 9

So the answer is just pi?

Answer 10

There is another solution or two coming from this though,\[\large\rm \cos x=0\]

Answer 11

No... \(\large\rm (\cos x-1)=0\) gave us a solution.
Read again what I posted if you're not sure.

\(\large\rm (\cos x+1)=0\) gave us another solution.

And \(\large\rm \cos x=0\) will give us two more solutions.

Answer 12

So we should end up with 4 in total.

Answer 13

3pi/2 ?

Answer 14

Is that one of the solutions?

Answer 15

Maybe, where did you get that from? :d

Answer 16

I am looking at the unit circle and cos is 3pi/2 at (0,-1)

Answer 17

So I know that pi and 3pi/2 are solutions. But I am confused how to get the other two, since you said there are 4 in all

Answer 18

cosine is your x-coordinate.
So at 3pi/2 you're getting the 0 from that coordinate pair.

Ok good, so that's one of our solutions!

Answer 19

So we want anywhere that cosine(the x-coordinate) is giving us zero. That would include up here at this point, ya?