Method for finding principal value:
Assume that we have to find principal value of θ satisfying the equation sinθ = - 1/2.
As sinθ is negative, θ will be in 3rd or 4th quadrant. We can approach 3rd or 4th quadrant from 2 directions. If we take anticlockwise direction numerical value of angle will be greater than Π. If we approach it in clockwise direction angle will be numerically less than Π. For the principal value, we have to take smallest angle numerically.
So for principal value :
1. If the angle is in 1st or 2nd quadrant we should select anticlockwise direction and if angle is in 3rd or 4th quadrant, we should select clockwise direction.
2. Principal value is never greater numerically than Π.
3. Principal value always lies in the 1st circle (that is in 1st rotation)
On above criteria θ will be or -Π/6. Among these 2 (-Π/6) has the least numerical value. Thus -Π/6 is the principal value of θ satisfying equation sinθ = -1/2.
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