Question

Find all solutions to the equation.

cos^2x + 2 cos x + 1 = 0

Answer 1

do you know how to factor a quadratic equation like :
\[t^2+2t+1=0\]

?

Answer 2

I know how to factor but I tried factoring that specific equation and couldn't figure it out. I'm not sure if I'm just blanking out or what.

Answer 3

So you do know factoring quadratics, nice..

Answer 4

Firstly, keep in mind, the expression \(\cos^2x\) means \((\cos x)^2\)

Answer 5

\[\cos^2x+2\cos x+1=0\]

is same as

\[(\cos x)^2+2\cos x+1=0\]

Answer 6

Now let \(\cos x = t\), the equation becomes :
\[t^2+2t+1=0\]


see if you can factor that

Answer 7

(t+1)^2?

Answer 8

Yes! it is an equation :
\[(t+1)^2=0\]

Answer 9

So t = -1?

Answer 10

Excellent!

Answer 11

but what is "t" ?

Answer 12

the given question has no "t"... we have introduced "t", so we need to get rid of it

Answer 13

t = -1

replace "t" by "cosx"

Answer 14

cos x = -1

Answer 15

Yes, for what angle x the cosine spits out -1 ?

Answer 16

180?

Answer 17

Yep! also remember that cosine repeats itself every 360 degrees...

Answer 18

therefore adding or subtracting 360 from a solution is also a solution :

\[180,~~180+360,~~180-360,~~\ldots \]

Answer 19

In compact form you may write the solutions as :
\[180 + 360*n\]

where \(n \) is all integers

Answer 20

That makes sense but I'm kind of confused as to what to do now.

Answer 21

we're done !

Answer 22

We are asked to find all the solutions to the given equation and we have found them.

Answer 23

Wouldn't I have to write what the specific solutions are? Like for 180 wouldn't it be pi?

Answer 24

180 degrees is same as pi radians

it doesn't matter what units you use

Answer 25

May I know how tall are you ?

Answer 26

5 ft 4

Answer 27

we don't use feet, can you tell me in meters please

Answer 28

I think its 1.6 meters

Answer 29

Would you agree that 5ft 4 inch is same as 1.6 meters ?

Answer 30

feet and meter are two different ways to express your height

your height is still the same, it doesn't change based on what units you choose to express it

Answer 31

Oh I see, radians and degrees are the same thing I was just getting confused because all the other answers on my homework are in radians.

Answer 32

analogously, degrees and radians are two different units for expressing measure of an angle

Answer 33

Yes, it seems that your textbook uses radians, so you better express your answer in radians

Answer 34

use below :

180 degrees = pi radians

Answer 35

180 + 360*n degrees

is same as

pi + 2pi*n radians

Answer 36

So pi, -pi, and 2pi radians would be the final answers?

Answer 37

Nope, they are not all the solutions

Answer 38

The solutions to the given equation are INFINITELY MANY

Answer 39

pi + 2pi*n

put n = 0, 1, 2, 3, ....

Answer 40

you get

pi, 3pi, 5pi, 7pi, ....


these all are solutions

Answer 41

Oh you could also plugin negative integers for "n"

Answer 42

pi + 2pi*n

put n = -1, -2, -3, ....

Answer 43

you get

-pi, -3pi, -5pi, ...

these are also solutions

Answer 44

as you can see, there are MANY MANY solutions, you cannot list them all in your notebook

Answer 45

Oh okay, that makes sense. Thank you for all your help!

Answer 46

np