We will discuss about the general solution of the equation 2 sin x minus 1 equals 0 (i.e., 2 sin x  1 = 0) or sin x equals half (i.e., sin x = ½).
How to find the general solution of the trigonometric equation sin x = ½ or 2 sin x  1 = 0?
Solution:
We have,
2 sin x  1 = 0
⇒ sin x = ½
⇒ sin x = sin \(\frac{π}{6}\)
⇒ sin x = sin (π  \(\frac{π}{6}\))
⇒ sin x = sin \(\frac{5π}{6}\)
Let O be the center of a unit circle. We know that in unit circle, the length of the circumference is 2π.
If we started from A and moves in anticlockwise direction then at the points A, B, A', B' and A, the arc length travelled are 0, \(\frac{π}{2}\), π, \(\frac{3π}{2}\), and 2π.
Therefore, from the above unit circle it is clear that the final arm OP of the angle x lies either in the first or in the second.
If the final arm OP of the unit circle lies in the first quadrant, then
sin x = ½
⇒ sin x = sin \(\frac{π}{6}\)
⇒ sin x = sin (2nπ + \(\frac{π}{6}\)), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)
Therefore, x = 2nπ + \(\frac{π}{6}\) …………….. (i)
Again, if the final arm OP of the unit circle lies in the second quadrant, then
sin x = ½
⇒ sin x = sin \(\frac{5π}{6}\)
⇒ sin x = sin (2nπ + \(\frac{5π}{6}\)), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)
Therefore, x = 2nπ + \(\frac{5π}{6}\) …………….. (ii)
Therefore, the general solution of equation sin x = ½ or 2 sin x  1 = 0 are the infinite sets of value of x given in (i) and (ii).
Hence general solution of 2 sin x  1 = 0 is x = nπ + (1)\(^{2}\) \(\frac{π}{6}\), n ∈ I
11 and 12 Grade Math
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