Question

Show how to solve (for blondes): 2 sin x + sqrt(3) = 0. Answers are 4pi/3 and 5pi/3.

Answer 1

2 sin x + sqrt(3) = 0
2sinx = - sqrt(3)
sinx = - sqrt(3) /2

For sinx = sqrt(3) /2
x = 60 degree = pi /3
Since it has a negative value, the angle is in quadrant III or quadrant IV
Therefore, x = pi + pi /3 = 4pi /3
and x = 2pi - pi /3 = 5pi /3

Answer 2

\[ \sin(x) = -\frac {\sqrt 3} 2= \sin(-\pi/3)
\]

Answer 3

\[ x = -\pi/3 + 2 k \pi \]
or
\[ x = \pi -(-\pi/3) + 2 k \pi = 5 \pi/3 +2 k \pi\]

Answer 4

\[ k = \pm 1, \pm 2, \pm 3, \cdots , \]