SOLUTION: Solve the following equation for the principal values:
2cos^3 x+2cos^2 x-cosx-1=0
Algebra.Com
Question 611643: Solve the following equation for the principal values:
2cos^3 x+2cos^2 x-cosx-1=0
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
I'm going to presume you mean "for the principal values of the inverse cosine"
\ +\ 2\cos^2(x)\ -\ \cos(x)\ -\ 1\ =\ 0)
This is much easier to see if you use substitution. Let )
Rearrange terms:
Factor a 
from the first two terms and a 1 from the second pair of terms:
\ +\ \left(2u^2\ -\ 1\right)\ =\ 0)
Then factor out 
\left(2u^2\ -\ 1\right)\ =\ 0)
Hence
\ =\ -1)
\ =\ \pm\frac{\sqrt{2}}{2})
The principal values of )
are in the closed interval 
Since \ \geq\ 0\ \forall\,x\,\in\ -\frac{\pi}{2}\ \leq\ z\ \leq\ \frac{\pi}{2})
Discard both negative roots. Use the unit circle to find the two values of 
in the principal interval that satisfy the equation. Hint: cosine is the coordinate of a point on the unit circle. Anothe hint: Angle values in the 4th quadrant from the point of view of the pricipal interval are negative values.
John
My calculator said it, I believe it, that settles it
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