Like cos2x = sinx.
is there a way to find it out WITHOUT graphing it?
is there a way to find it out WITHOUT graphing it?
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Use the identity
cos 2x = 1 - 2 (sinx)^2
So your problem will be a quadratic in sinx:
1- 2(sinx)^2 = sinx
=> 2(sinx)^2 +sinx-1 = 0
From the quadratic formula:
sinx = -1/4 + sqrt(1+8)/4 or sinx = -1/4 - sqrt(1+8)/4
i.e. sinx = -1/4 + 3/4 or sinx = -1/4-3/4
sinx lies between -1 and 1 inclusive, so both roots are allowed
Thus the solutions are:
sinx = -1/4 + 3/4 = 1/2
or
sinx = -1/4-3/4 = -1
you can then sove each of these for x
cos 2x = 1 - 2 (sinx)^2
So your problem will be a quadratic in sinx:
1- 2(sinx)^2 = sinx
=> 2(sinx)^2 +sinx-1 = 0
From the quadratic formula:
sinx = -1/4 + sqrt(1+8)/4 or sinx = -1/4 - sqrt(1+8)/4
i.e. sinx = -1/4 + 3/4 or sinx = -1/4-3/4
sinx lies between -1 and 1 inclusive, so both roots are allowed
Thus the solutions are:
sinx = -1/4 + 3/4 = 1/2
or
sinx = -1/4-3/4 = -1
you can then sove each of these for x