alexh107
  • alexh107
Find all solutions to the equation. cos^2x + 2 cos x + 1 = 0
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
ganeshie8
  • ganeshie8
do you know how to factor a quadratic equation like : \[t^2+2t+1=0\] ?
alexh107
  • alexh107
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.
ganeshie8
  • ganeshie8
So you do know factoring quadratics, nice..

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

ganeshie8
  • ganeshie8
Firstly, keep in mind, the expression \(\cos^2x\) means \((\cos x)^2\)
ganeshie8
  • ganeshie8
\[\cos^2x+2\cos x+1=0\] is same as \[(\cos x)^2+2\cos x+1=0\]
ganeshie8
  • ganeshie8
Now let \(\cos x = t\), the equation becomes : \[t^2+2t+1=0\] see if you can factor that
alexh107
  • alexh107
(t+1)^2?
ganeshie8
  • ganeshie8
Yes! it is an equation : \[(t+1)^2=0\]
alexh107
  • alexh107
So t = -1?
ganeshie8
  • ganeshie8
Excellent!
ganeshie8
  • ganeshie8
but what is "t" ?
ganeshie8
  • ganeshie8
the given question has no "t"... we have introduced "t", so we need to get rid of it
ganeshie8
  • ganeshie8
t = -1 replace "t" by "cosx"
alexh107
  • alexh107
cos x = -1
ganeshie8
  • ganeshie8
Yes, for what angle x the cosine spits out -1 ?
alexh107
  • alexh107
180?
ganeshie8
  • ganeshie8
Yep! also remember that cosine repeats itself every 360 degrees...
ganeshie8
  • ganeshie8
therefore adding or subtracting 360 from a solution is also a solution : \[180,~~180+360,~~180-360,~~\ldots \]
ganeshie8
  • ganeshie8
In compact form you may write the solutions as : \[180 + 360*n\] where \(n \) is all integers
alexh107
  • alexh107
That makes sense but I'm kind of confused as to what to do now.
ganeshie8
  • ganeshie8
we're done !
ganeshie8
  • ganeshie8
We are asked to find all the solutions to the given equation and we have found them.
alexh107
  • alexh107
Wouldn't I have to write what the specific solutions are? Like for 180 wouldn't it be pi?
ganeshie8
  • ganeshie8
180 degrees is same as pi radians it doesn't matter what units you use
ganeshie8
  • ganeshie8
May I know how tall are you ?
alexh107
  • alexh107
5 ft 4
ganeshie8
  • ganeshie8
we don't use feet, can you tell me in meters please
alexh107
  • alexh107
I think its 1.6 meters
ganeshie8
  • ganeshie8
Would you agree that 5ft 4 inch is same as 1.6 meters ?
ganeshie8
  • ganeshie8
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
alexh107
  • alexh107
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.
ganeshie8
  • ganeshie8
analogously, degrees and radians are two different units for expressing measure of an angle
ganeshie8
  • ganeshie8
Yes, it seems that your textbook uses radians, so you better express your answer in radians
ganeshie8
  • ganeshie8
use below : 180 degrees = pi radians
ganeshie8
  • ganeshie8
180 + 360*n degrees is same as pi + 2pi*n radians
alexh107
  • alexh107
So pi, -pi, and 2pi radians would be the final answers?
ganeshie8
  • ganeshie8
Nope, they are not all the solutions
ganeshie8
  • ganeshie8
The solutions to the given equation are INFINITELY MANY
ganeshie8
  • ganeshie8
pi + 2pi*n put n = 0, 1, 2, 3, ....
ganeshie8
  • ganeshie8
you get pi, 3pi, 5pi, 7pi, .... these all are solutions
ganeshie8
  • ganeshie8
Oh you could also plugin negative integers for "n"
ganeshie8
  • ganeshie8
pi + 2pi*n put n = -1, -2, -3, ....
ganeshie8
  • ganeshie8
you get -pi, -3pi, -5pi, ... these are also solutions
ganeshie8
  • ganeshie8
as you can see, there are MANY MANY solutions, you cannot list them all in your notebook
alexh107
  • alexh107
Oh okay, that makes sense. Thank you for all your help!
ganeshie8
  • ganeshie8
np

Looking for something else?

Not the answer you are looking for? Search for more explanations.