Question

Help?
Find all solutions to the equation in the interval [0, 2π).

cos 2x - cos x = 0

Answer 1

So we could express this in a quadratic like form.
Use the identity:
\[**\cos(2x)=2\cos^2(x)-1 \]




This is how I got it:

\[\cos(2x)=\cos^2(x)-\sin^2(x)=\cos^2(x)-(1-\cos^2(x))=2\cos^2(x)-1 \]

Answer 2

So you are really asked to solve
\[2\cos^2(x)-\cos(x)-1=0\]
And you should know how to solve
\[2u^2-u-1=0\]

Answer 3

So I factor it?
@myininaya

Answer 4

Yep or quadratic formula.

Answer 5

Awesome.

Answer 6

Hey... I got -1/2, and 1, but how do I get the answers like my choices? They have pi and such in them.. I don't get it..

Answer 7

ok so you solve for cos(x)

Answer 8

now you need to solve for x

Answer 9

\[\cos(x)=\frac{-1}{2} \text{ or } \cos(x)=1 \]

Answer 10

Solve both of those equations

Answer 11

Hi, BatGirl:
If you have a TI-84 calculator, you could type in
-1
cos 1, which would result in your obtaining the angle x = 0.

Or you could graph y=cos x on the given interval, and note from that x=0 is the only x value on that interval at which cos x = 1.

Answer 12

Wait, I could type in what? @mathmale

Answer 13

Or you could use the unit circle.

Answer 14

http://en.wikipedia.org/wiki/Unit_circle

Answer 15

\[A) 0, 2\pi/3, 4\pi/3\]
\[B) \pi/6, 5\pi/6, 3\pi/2\]
\[C) 0, \pi/2, 7\pi/6, 11\pi/6\]
D) No solution

Answer 16

For what angles on your unit circle is the cosine value 1 and -1/2?

Answer 17

Just like the x coordinates on your table/unit circle

Answer 18

pi/3, and pi/2?

Answer 19

at pi/2 cos is 0
at pi/3 cos is 1/2

Answer 20

Ugh. I'm not very good with this circle thing..

Answer 21

I'm asking you to look at that circle and tell me for what angles is the x-coordinate -1/2 or 1

Answer 22

So it's 0, and..

Answer 23

5pi/3?

Answer 24

Look to see where you have the x-coordinate is 1 first?
Can you name these values?
yes at 0 we have cos is 1
now also when is the x-coordinate -1/2?

Answer 25

Look at the x-coordinates until you see -1/2 you should see and angle near by corresponding to that point
I'm asking you for that angle

Answer 26

There is actually two angles in between 0 and 2pi so that cos value is -1/2

Answer 27

300?

Answer 28

that is 1/2 not -1/2

Answer 29

So looking at that circle you don't see any x-coordinates being -1/2?
Do you know which one is the x-coordinate?
The very first value of any rectangular point is the x-value as long as we are treating the horizontal axis as the x-axis which is what we normally do.

Answer 30

Oh! Oops! 240.

Answer 31

which 4pi/3 in radians
now one more angle will give you the cos is -1/2

Answer 32

hint look in the second quadrant

Answer 33

120, or 2pi/3. But there's another one.. How do we know to only use those two?

Answer 34

So we know to use 0, 4pi/3, and 2pi/3 because those are the only angles between 0 and 2pi that give us cos is either 1 or -1/2

Answer 35

Wait, nevermind, that one is a y-value. My bad.

Answer 36

What does this mean? Rewrite with only sin x and cos x.

cos 2x + sin x

Answer 37

Well your expression is in terms of cos(2x) and sin(x)
not sin(x) and cos(x)

Answer 38

You have to use an identity to rewrite cos(2x)

Answer 39

the same identity I gave you above.

Answer 40

That's what I thought, but none of my answers have a cos in them... :c

Answer 41

Well rewrite it using sine then

Answer 42

you know cos^2(x)=1-sin^2(x)

Answer 43

It's not squared, just has a 2x though.

Answer 44

? we said earlier cos(2x)=2cos^2(x)-1
I'm saying use this identity cos^2(x)=1-sin^2(x) to rewrite cos(2x) in terms of sin

Answer 45

Oh.. My bad again. I think I can get that. Thanks so much! God bless you!

Answer 46

Hi, BatGirl:
If you have a TI-84 calculator, you could type in
-1
cos 1, which would result in your obtaining the angle x = 0.

By that I meant the inverse cosine of 1.
-1
If you are told that the cosine of a certain angle is 1, then cos 1 on your calculator will produce the angle in either degrees or radians, depending on the MODE you've selected.