Solving Trigonometric Equation: Alternate Methods?

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The discussion revolves around solving the trigonometric equation 2 cos 2x - 4 cos x + 3 = 0 within the interval 0 < x < 2π. The user attempts to simplify the equation using the identity for cos 2x, leading to a quadratic form that is challenging to factor. After some manipulation, they arrive at (2cos x - 1)² = 0, which yields solutions of cos x = 1/2, corresponding to x = 60° or 300°. The user expresses frustration over the complexity of the problem but ultimately finds a solution. The conversation touches on the challenges of returning to trigonometry after a break, highlighting the importance of practice and familiarity with identities.
Short term memory
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ok here's the problem i haven't taken trig for like 2 years and suddenly i have been thrown back into it...

problem: 2 cos 2x - 4 cos x + 3 = 0 0 < x < 2 pi

my work (used cos 2 x = 2 cos^2 x - 1)

4 cos^2 x - 2 - 4 cos x + 3 = 0

4 cos^2 x - 4 cos x + 1 = 0

ok now here is my problem if i try and factor it out i ll be left with the a really ugly 1/ ( 4 cos x)

is there another equation i could use... is my logic flawed... is there an easier way... do i just have to do it the ugly way... ?
 
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2cos2x - 4cosx + 3 = 0
2(2cos^{2}x-1) - 4cosx + 3 = 0
4cos^{2}x - 4cosx + 1 = 0
(2cosx-1)^{2} = 0
cosx = 1\2
x = 60 deg. or 300 deg.
 
3.14159265358979 said:
2cos2x - 4cosx + 3 = 0
2(2cos^{2}x-1) - 4cosx + 3 = 0
4cos^{2}x - 4cosx + 1 = 0
(2cosx-1)^{2} = 0
cosx = 1\2
x = 60 deg. or 300 deg.

... me stupid...

PS having a little to much fun with pi arn't we... you should go out to atleast when pi gets to 123456789...
 
Short term memory said:
PS having a little to much fun with pi arn't we... you should go out to atleast when pi gets to 123456789...


"There is not enough room in this margin..." :wink:
 

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