Question

Find all values of x in the interval [0, 2π] that satisfy the equation. (Enter your answers as a comma-separated list.)
18 + 9 cos(2x) = 27 cos(x)

Answer 1

hint subtract 27cosx from both side

Answer 2

i bring to a point where it is easy to do
9cos2x-27cosx+18=0
cos2x-3cosx+2=0

Answer 3

cos2x=2cos^2x-1

Answer 4

so we have 2cos^2x-3cosx+1=0

Answer 5

surely you can do an equation like this 2x^2-3x+1=0

Answer 6

can you?

Answer 7

Aha, Thank you very much for replying. Let me try it and get back to you

Answer 8

ok

Answer 9

I get it until 2cos^2x-3cosx+1=0

Answer 10

alright
now we just think of that as 2r^2-3r+1=0
for now we forgot that cosx like we replaced with r

Answer 11

okay...

Answer 12

2r^2-3r+1=0 is just quadratic equation

Answer 13

ohh I see...

Answer 14

okay so r = 1/2 or 1 yeah?

Answer 15

you solve for r just as you usually do
-factoring
-quadratic formula
-completing the square
i don't know what other stuff you can use lol

Answer 16

yes seems good

Answer 17

lol...okay, so replacing r with cos will result in...?

Answer 18

if i did factoring we get (2r-1)(r-1)=0
which agrees with what you got
r=1/2 and r=1
we replace r by cosx
cosx=1/2 and cosx=1

Answer 19

now you think of all possible angles that give 1/2 and 1
from 0 to 2pi

Answer 20

forr what angles do we have cos =1/2?

Answer 21

60

Answer 22

yes

Answer 23

and cos 0 = 1

Answer 24

not onnly 0 though

Answer 25

360 too

Answer 26

in radians that will be pi/3...I mean for 60...yeah?

Answer 27

yeah

Answer 28

yeah...360 too

Answer 29

360 or 2pi in radian

Answer 30

that will be 2pi

Answer 31

yeah

Answer 32

cos 300...5pi/3

Answer 33

then x=0,pi/3,2pi

Answer 34

that gives 1/2 yeah?

Answer 35

we only need values from 0 to 2pi and we stop
we don;t need other rotations

Answer 36

yeah...but 5pi/3 is actually smaller than 2pi

Answer 37

i was giving you how to put it as a complete answer with commas as they asked
x=0,pi/3, 2pi

Answer 38

oh yeah right that has cos =1/2 too
correct i missed that one

Answer 39

good catch

Answer 40

Yeah

Answer 41

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Answer 42

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Answer 43

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Answer 44

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Answer 45

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Answer 46

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Answer 47

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Answer 48

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