Question

solve 2cos^2(3x)+3cos(3x)+1=0 for x=[0,2pi)

Answer 1

you have a quadratic in cosine

Answer 2

if we said let u=cos3x. WHat would your eq become?

Answer 3

it would be 2cos^2(u)+3cos(u)+1=0

Answer 4

not quite \[\color\red{cos 3x}=u\]

Answer 5

oh 2u^2+3+1=0

Answer 6

well 3u but very close

Answer 7

omg, im sorry.

Answer 8

2u^2+3u+1=0

Answer 9

it's cool , ok so now could you solve that for u?

Answer 10

and don't worry, first time I saw one of these god awful things I was beyond clueless

Answer 11

1/2 or -1/2 ?

Answer 12

wolfram agrees! http://www.wolframalpha.com/input/?i=solve+2u%5E2%2B3u%2B1%3D0

Answer 13

ok so now, you need to substitute your u value back in and solve both outcomes

Answer 14

And when it equals 0 it is that one? so it would be -1/2?

Answer 15

wait wheredid that come from?

Answer 16

i plugged in (-1/2) and (1/2) for U and then i got 0 and 3 when plugged in

Answer 17

it's just what was the substitution we had for u?

Answer 18

u is cos3x

Answer 19

so would it then be 2(-1/2)^2+3(-1/2)+1 ?

Answer 20

uhm, just wait a sec, so u=cos3x yeah? u also equals 1/2 and -1/2

Answer 21

you shouldn't get zero ever. Can you show me your work?

Answer 22

ok that's why, you should solve cos3x=1/2 and cos3x=-1/2

Answer 23

i think i am still doing it wrong because then it would be cos3x= 60 degrees or 120 degrees? and then do you divide that by 3 so then it would be 20 degrees or 40 degrees?

Answer 24

no, ok so if you have 5x= 4 what do you do?

Answer 25

its x=4/5

Answer 26

ok so now you have cos(5x)=4, what do you have to do first?

Answer 27

you can't just divide the 5

Answer 28

cos^-1(4)=5x?

Answer 29

then divide by..5?

Answer 30

Heloo new person..?

Answer 31

Let's wait for _lostmathstudent to figure it out

Answer 32

but thats what i did

Answer 33

i did the inverse and got 60 for the 1/2

Answer 34

and then so for the -1/2 it could be either 120 or 240 or 300?

Answer 35

by you then get 3x=60. not just 60

Answer 36

so then i'd divide all of them by 3 and then get 20, 40, 80 and 100?

Answer 37

wait wait wait

Answer 38

okay sorry

Answer 39

you are so not done with the first part, one step at a time here.

Answer 40

\[cos(3x)=.5\]
\[cos^{-1}[cos(3x)]=cos^{-1}(.5)\]
Do you follow?

Answer 41

okay i follow.

Answer 42

so now, we simplify yes? so when is the cosine positive?

Answer 43

in q1

Answer 44

anywhere else?

Answer 45

and q4

Answer 46

yep ok so now, what is the value in 1 and what is the value in 4 of arccos(.5)?

Answer 47

it'd be 60 and 300?

Answer 48

yep, now, you have two eqs. 3x=60 and 3x=300

Answer 49

so then its 20 or 100?

Answer 50

both, you will have 4 answers here

Answer 51

so now, you repeat for -.5

Answer 52

so its 20 and 100 and then for the negative cosine it would be 120 and 240 and so then the final 4 answers would be 20,100, 40, and 80?

Answer 53

sounds right

Answer 54

awesome thank you !

Answer 55

np, do you understand now?

Answer 56

yes i do :)

Answer 57

yay!