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