How do you find the general solutions for #sin(2a)-cos(a)-2sin(a)+1=0#?

1 Answer
Nghi N.
Oct 22, 2015

Solve sin 2x - cos x - 2sin x + 1 = 0

Ans: #2pi; pi/6, and (5pi)/6#

Explanation:

Replace in the equation sin 2x by (2sin x.cos x), we get:
2sin x.cos x - cos x - 2sin x + 1 = 0
Put 2sin x in common factor:
2sin x(cos x - 1) - (cos x - 1) = 0
(cos x - 1)(2sin x - 1) = 0

a. cos x - 1 = 0 --> cos x = 1 --> x = 0 and #x = 2pi#
b. 2sin x - 1 = 0 --> #sin x = 1/2# --> #x = pi/6# and #x = (5pi)/6#
General solution:
#x = 2kpi#
#x = pi/6 + 2kpi#
#x = (5pi)/6 + 2kpi#

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