Question

Solve this equation: (1 - 2sinx)cos x/ [(1 + 2 sinx)(1-sinx)] = Squrt(3)

Answer 1

wow. tough. i see no easy way of solving this.

substitute y = sin(x) and you get 8y^3 - 6y - 1 = 0

there are no rational solutions to this cubic equation.

it is possible to find exact solutions, but that would be a horrible, horrible journey to go on.

Answer 2

I know this is a tough one :)

Answer 3

do you know the answer? i'd be interested to know

Answer 4

Yeah i have the answer, but just wanted to know if anyone have a better way to solve :P

Answer 5

how did you solve it?

Answer 6

cos x - 2 sinx cosx= squrt(3) (1 - sinx + 2sinx - 2 (sinx)^2)
\[\cos x - \sin2x = \sqrt{3} - \sqrt{3}sinx + 2\sqrt{3}sinx - \sqrt{3}(1 - \cos2x) \]
\[cosx - \sin2x = \sqrt{3} - \sqrt{3}sinx + 2\sqrt{3}sinx - \sqrt{3} + \sqrt{3}\cos2x\]
\[cosx - \sin2x = \sqrt{3}sinx + \sqrt{3}\cos2x\]
\[cosx - \sqrt{3}sinx = \sin2x + \sqrt{3}\cos2x\]
Then use the sum of difference formula:
\[(cosx)/2 -( \sqrt{3}/2)sinx = (\sin2x)/2 + (\sqrt{3}/2)\cos2x\]
\[cosx \cos \pi/3 - sinx \sin \pi/3 = \sin2xsin \pi/6 + \cos 2xcos \pi/6\]
\[\cos (x + \pi/3) = \cos (2x - \pi/6)\]
\[x + \pi/3 = 2x - \pi/6 => x = \pi/2 + k2\pi\]

Answer 7

but from that part, they got another answer is x = -pi/18 + k2pi/3

and in the condition in the beginning sinx can't be 1 or -1/2
therefore the final answer is x= -pi/18 +k2pi/3


Well, i understood the whole steps...but except the last answer, im not sure yet how they get x= -pi/18 +k2pi/3

Answer 8

that's brilliant! i didn't even think of compound angle formulas.

here's how they got the answer:

cos(x + pi/3) = cos(2x - pi/6)

1. x + pi/3 = 2x - pi/6 + 2*pi*k

or

2. x + pi/3 = -(2x - pi/6) + 2*pi*k

As you said, 1 ends up with sin(x) = 1, which is not allowed.

So we go with 2:

x + pi/3 = -2x + pi/6 + 2*pi*k

so 3x = pi/6 - pi/3 + 2*pi*k

so 3x = -pi/6 + 2*pi*k

so x = -pi/18 + 2*pi*k/3

Answer 9

oh I see now, my bad I didn't think about the other case :P But you saw that...
Yeah, this question looks simple at first but give me a big headache later :D