Question

Free medal given!! HELP!
solve each equation on the interval [0,2pi)
sin3x=-1/2

Answer 1

@bibby

Answer 2

@misty1212

Answer 3

2cos^2 x + 5cos x + 2 = (2cos x + 1)(cos x + 2) = 0
cos x = -1/2
x = 2pi/3, or 4pi/3

Answer 4

i'm talking about for sine

Answer 5

@nikato

Answer 6

so you know how to solve sin(u)=-1/2
use the unit circle

Answer 7

by the way if x is between 0 and 2pi
so if u=3x then u/3=x
so u/3 is between 0 and 2 pi
therefore u is between 0(3) and 2pi(3)
or if we simplify that u is between 0 and 6pi
so solve sin(u)=-1/2 on the interval [0,6pi] first
then we will solve for x

Answer 8

x will be 4pi/3 and 5pi/3

Answer 9

what did you solve?

Answer 10

are you solving this for u first:
sin(u)=-1/2 on [0,6pi]?

Answer 11

i dont understand how to do that. i'm just diong what i know. the question is sin3x=-1/2 i want to get the steps in the fastest quickest way popssible

Answer 12

so have you ever used the unit circle?
because that is normally how you solve something like sin(u)=-1/2 on the interval I gave

Answer 13

you find for what angles you have the y value is -1/2

Answer 14

i have but im not really sure. can you show me the steps you will take

Answer 15

for example I will find one value for you
\[\sin(u)=\frac{-1}{2} \\ \text{ when } u=\frac{7 \pi}{6}\]
can you find any other values where the angle gives you the y value is -1/2

Answer 16

all you are doing is using a table

Answer 17

look at the unit circle and just observe when the y-coordinate is -1/2

Answer 18

hint: look all in the 4th quadrant

Answer 19

also*

Answer 20

11pi/6

Answer 21

yes

Answer 22

there are some more values for u
since we are looking from u=0 to u=6pi

Answer 23

no it stops at 2pi

Answer 24

\[u=\frac{7\pi}{6}, 2\pi+\frac{7\pi}{6},4\pi+\frac{7\pi}{6} \\ u=\frac{11\pi}{6},2\pi+\frac{11\pi}{6},4\pi+\frac{11\pi}{6}\]
now you can finally solve for x
replace u with 3x

Answer 25

and solve those 6 equations

Answer 26

as i said earlier x is between 0 and 2pi
but we replaced 3x with u
which means u=3x and x=u/3
so u/3 is between 0 and 2pi
so u is actually between 0 and 6pi
so we had to solve sin(u)=-1/2 on the interval [0,6pi]

Answer 27

i dont understand can yuou explain it to me in equation form?

Answer 28

do you know how to replace u with 3x?

Answer 29

i would go about this by doing 3x=4pi/3+2npi

Answer 30

\[3x=\frac{7\pi}{6},2\pi+\frac{7\pi}{6},4\pi+\frac{7\pi}{6} \\ 3x=\frac{11\pi}{6},2\pi+\frac{11\pi}{6},4\pi+\frac{11\pi}{6}\]

Answer 31

now solve for x

Answer 32

the explain it ot me b making equations first sin is negative in the 3rd and 4th quariants and isnt that 4pi3 and 5pi/3

Answer 33

are you chaging the equation to sin(x)=-sqrt(3)/2 ?

Answer 34

no

Answer 35

then how are you getting those values

Answer 36

i don tknow if its right. i was asking you

Answer 37

so you want to resolve sin(u)=-1/2 then?

Answer 38

again we are looking for angle u such that the y-coordinate is -1/2

Answer 39

we discovered the values 7pi/6 and 11pi/6 satisfy this

Answer 40

then we had u was between 0 and 6pi
so that means we have three rotation of answers for each

Answer 41

so we have 6 answers in all

Answer 42

which were written above

Answer 43

but now you have to solve for x

Answer 44

by dividing 3 on both sides

Answer 45

\[x=\frac{7\pi}{6(3))},\frac{2\pi}{3}+\frac{7\pi}{6(3)},\frac{4\pi}{3}+\frac{7\pi}{6(3)} \\ x=\frac{11\pi}{6(3)},\frac{2\pi}{3}+\frac{11\pi}{6(3)},\frac{4\pi}{3}+\frac{11\pi}{6(3)} \\ \]

Answer 46

I think you still don't get it.
I will try to explain one more time.
sin(3x)=-1/2 where 0<=x<=2pi
Let 3x=u. We are going to put everything in terms of u including the inequality.
sin(u)=-1/2 where 0<=u/3<=2pi since u=3x => u/3=x
solving the inequality gives 0<=u<=6pi
so we need to solve sin(u)=-1/2 on [0,6pi]
So in the first period [0pi,2pi] we have u=7pi/6 or u=11pi/6
in the second period [2pi,4pi] we have u=2pi+(7pi/6) or u=2pi+(11pi/6)
in the third period [4pi,6pi] we have u=4pi+(7pi/6) or u=4pi+(11pi/6)

now we need to remember our actual objective
solve for x
recall we said u IS 3x
so in all of these equations replace u with 3x since we made u equal 3x

so you have the following 6 equations:
3x=7pi/6
3x=11pi/6
3x=2pi+(7pi/6)
3x=2pi+(11pi/6)
3x=4pi+(7pi/6)
3x=4pi+(11pi/6)

now solve for x by dividing each of them by 3.