Question

How do you solve `3sinx=sinx-1`?

Answer 1

`x= -pi/6 + 2pi n` or `x=(7pi)/6 + 2pin` `{n in ZZ}`

Explanation:

`3sin x = sin x -1`

`2sinx =-1`

`sinx=-1/2`

`x = arcsin(-1/2)`

`x = -pi/6` for `x in (-pi,pi) or x=(7pi)/6` for `x in (pi, 2pi)`

In general: `x= -pi/6 + 2pi n` or `x=(7pi)/6 + 2pin` `{n in ZZ}`
Since the period of the `sin` function is `2pi`

Answer 2

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

Explanation:

3sin x = sin x - 1
2sin x = -1 --> `sin x = - 1/2`
Trig table of special arcs -->
`sin x = - 1/2` --> arc `x = - pi/6` or `x = (11pi)/6` (co-terminal)
Trig unit circle gives another arc that has the same sine value (-1/2):
arc `x = (7pi)/6`
Answers for `(0, 2pi)`:
`(7pi)/6 and (11pi)/6`
General answers:
`x = (7pi)/6 + 2kpi`
`x = (11pi)/6 + 2kpi`