Question

how to solve cos (4x)=-1/2 ?

Answer 1

find a number (angle if you like) whose cosine is \(-\frac{1}{2}\) which should not be too hard

Answer 2

if it is not obvious, look at the unit circle on the last page of the attached cheat sheet
see what angle has a first coordinate of \(-\frac{1}{2}\)

Answer 3

wouldnt that be both pi/3 and 5pi/3??? would this yield two general answers?

Answer 4

no \(\cos(\frac{\pi}{3})=\frac{1}{2}\) you want \(-\frac{1}{2}\)

Answer 5

try \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\)

Answer 6

this tells you
\[4x=\frac{2\pi}{3}\] and so
\[x=\frac{\pi}{6}\]

Answer 7

or
\[4x=\frac{4\pi}{3}\] and therefore \[x=\frac{\pi}{3}\]

Answer 8

oh ya! so then it still gives the answers of
2pi/3 +2pi n
and
4pi/3 +2pi n
?

Answer 9

wat about the other value of x?

Answer 10

no you are solving for \(x\) not for \(4x\)

Answer 11

gotta divide your answers by 4

Answer 12

\[4x=\frac{2\pi}{3}+2n\pi\]
\[x=\frac{\pi}{6}+\frac{1}{2}n\pi\]

Answer 13

I mean like cos(2pi/3) and cos(4pi/3)...

Answer 14

would u also plug in the 4pi/3, i mean.

Answer 15

hold on lets go slow

Answer 16

\[\cos (\frac{2\pi}{3})=-\frac{1}{2}\] right?

Answer 17

yes

Answer 18

and your job was to solve
\[\cos(4x)=-\frac{1}{2}\]

Answer 19

yes

Answer 20

this tells you that
\[4x=\frac{2\pi}{3}+2n\pi\]

Answer 21

yea

Answer 22

now divide by \(4\) to solve for \(x\) and get
\[x=\frac{\pi}{6}+\frac{1}{2}n\pi\]

Answer 23

or however you wish to write it

Answer 24

So my question is,

Answer 25

are there two answers, since there are two different angles that cosine can be if it's -1/2?

Answer 26

right
now repeat the process with
\[4x=\frac{4\pi}{3}+2n\pi\]

Answer 27

yes there are two answers, or rather two formulas for infinite number of answers

Answer 28

pi/6 + pi n/2
and
pi/3 + pi n/2

yes? :)

Answer 29

yes!

Answer 30

>o< ok then that clears things up~ and the +2pi n thing would be just +pi n for tangent and cotangent, right? I just need to clear that up...

Answer 31

yes

Answer 32

is there a plus or minus thing involved in solving trig equations? I went in to tutoring this morning and heard something about having to put plus or minus in the process, but I had to go early, so I wasn't able to ask what scenario that was in... I think I heard something about cosine...

Answer 33

hello?