SOLUTION: Solve the following equation for the principal values: 2cos^3 x+2cos^2 x-cosx-1=0

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Question 611643: Solve the following equation for the principal values:
2cos^3 x+2cos^2 x-cosx-1=0
Answer by solver91311(24713) About Me  (Show Source):
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I'm going to presume you mean "for the principal values of the inverse cosine"



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:



Then factor out



Hence





The principal values of are in the closed interval

Since

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

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