Question

Which is the correct answer `sinx < -1/2` ?

`sinx < -1/2`
1. Use formulas
`pi-arcsin(-1/2)+2pik`
`2pi+arcsin(-1/2)+2pik`
So
`(7pi)/6+2pik`
`(11pi)/6+2pik`
2. Use unit circle
And we get
`-(5pi)/6`
`-pi/6`
Is the same?

Answer 1

`x in (( 2 k - 1 )pi + 1/6 pi ), 2 k pi - 1/6 pi ), k = 0, +- 1, +- 2, +-3, ...` . I will give more details, later.

Explanation:

In `[ - pi, pi ],`

`sin x < -1 /2`, when `x in ( - 5/6 pi, - 1/6 pi ) in Q_3 and Q_4`

For `x in ( - oo, oo )`,

`sin x < -1 /2`, when

`x in (( 2 k - 1 )pi + 1/6 pi ), 2 k pi - 1/6 pi ), k = 0, +- 1, +- 2, +-3, ...`

`= ...U ( - 17/6 pi, -13/6 pi ) U ( -5/6 pi, - 1/6 pi )`

`U ( 7/6 pi, 11/6 pi ) U...`

See graph for `y < 0 in Q_3 and Q_4`.
graph{y-(sin x + 1/2)=0 [-10 10, -1 0]}