Question

How do I know when I use substitution when solving trig equations...?

Answer 1

Such as \[\sqrt{\cos x}=2\cos x-1\]

Answer 2

How do I know if I can just isolate cos or I have to make cos=to something and substitute?

Answer 3

you could probably try squaring both side there

Answer 4

you will end up with a quadratic in terms of cos(x)

Answer 5

you can decide to substitute u for cos(x) if it makes it less confusing to
you
but the sub is really not needed

Answer 6

quadratic.? and solve using quadratic formula or factor?

Answer 7

well if it factorable I recommend factoring
if not then yes quadratic formula

Answer 8

\[a \cdot \cos^2(x)+b \cdot \cos(x)+c=0 \\ \cos(x)=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

Answer 9

let me try both ways

Answer 10

k
it actually can be done both ways

Answer 11

I made cosx=x

square both sides and ended up with a quadratic anyways lol

Answer 12

4p^2-5p+1=0

p=1,1/4

Answer 13

which means that...

cosx=1/4
cosx=1

cosx=1 when it's 0 degrees

Answer 14

not sure about 1/4

Answer 15

are you only finding one solution per equation?

Answer 16

this would be the general solution

Answer 17

the solution for cos(x)=1 is x=2k*pi, where k is an integer

Answer 18

if this was a set domain of 0<=x<=2pi

it would be...90 and 180 degrees?

Answer 19

oh I wasn't given the domain
cos(x)=1 when x=0 or 2pi since cos(0)=1 and cos(2pi)=1

\[\cos(x)=\frac{1}{4} \text{ in the first rotation when } x=\arccos(\frac{1}{4}) \text{ and } x=-\arccos(\frac{1}{4})\]

Answer 20

there isn't a domain lol I was just asking IF there was one

Answer 21

I knew cos(x) is an even function
which means cos(x)=cos(-x)
so I solved cos(x)=1/4 and cos(-x)=1/4 using the arccos( ) function

Answer 22

1/4 is a solution? if you plug that back into the original equation you get

1/2=-1/2

is it not extraneous?

Answer 23

nope 1/4 isn't a solution

Answer 24

so general solution just

x=2npi

Answer 25

cos(x)=1 when x=2npi where n is integer
cos(x)=1/4 when x=arccos(1/4)+2npi or x=arccos(-1/4)+2npi
Bu what I think you meant to say any solutions to cos(x)=1/4
is not a solution to the original since 1/2 doesn't equal 1/2-1=-1/2

Answer 26

so yes the final solution is x=2npi

Answer 27

alright ty, i still confused on the approach of questions like these so im gonna post more questions

Answer 28

i get the idea but not how to isolate or substitute

Answer 29

practice practice
I don't think in math it is always easy to see how to solve an equation
that is why plenty of practice is what most teachers suggest ( I think they suggest that-lol; that is what I would suggest)