Question

find all solutions to the equation. cos2x + 2 cos x + 1 = 0

Answer 1

Is that meant to be cos^2(x) or actually cos2x?

Answer 2

cos ^2x

Answer 3

if you substitute u for cos x what is the equation?

Answer 4

Ah, okay. Well, start off by pretending that those aren't cosines, but actually x's and factor like a quadratic.

Answer 5

Replace cos(x) with u. Solve for u

Answer 6

So it would be u^2 +2u+1=0

Answer 7

Yep.

Answer 8

right now how does that factor?

Answer 9

u ( u+2) +1 = 0

Answer 10

would that be it

Answer 11

no...you can't factor only part of a polynomial. Trinomials usually factor to two binomials. Do you know how to factor trinomials?

Answer 12

I dont know i remembering it just forgot it

Answer 13

remember it

Answer 14

so x^2 + 2x + 1 ... I usually consider what could possibly multiply to give me x^2 (the first term) and then what could multiply to give me +1 (the last term) then I test all the possible combinations. In this case it is pretty easy because there is only one combinAtion.

Answer 15

x*x = x^2 and +1*+1=+1 or -1*-1 = +1. So it is either (x+1) (x+1) or (x-1)(x-1).

Answer 16

But (x-1)(x-1) = x^2-2x+1 and (x+1) (x+1) = x^2+2x+1 so it is (x+1) (x+1)

Answer 17

so since u^2+2u+1=(u+1)(u+1)=0 , you know either u+1 =0 or u+1 =0 which is redundant.

Answer 18

Sine u+1=0 then substitute cos x back in for u and solve from there.

Answer 19

I would strongly suggest reviewing factoring when you are studying this part of trigonometry. It will make your life much easier.

Answer 20

Ok thanks

Answer 21

Could you still help me on trying trying to find all solutions to it ?

Answer 22

Ok so you have that (u+1)(u+1)=0 ...so we are really looking for U+1=0.

Answer 23

SO this means that u= -1 but remember u = cos x so cos x = -1. Now based on what you know from the unit circle when does cos x = -1?

Answer 24

pi

Answer 25

yep but you have to consider all solutions so ever revolution that would land at that same spot so...
\[x=\pi + 2\pi n \]
where
\[n \in \mathbb{Z}\]
the line above is simply saying that n is an integer so that only when the value is pi or any revolution that lands on pi (i.e. all co terminal angles to pi)