4.cosec x=-2
Given that,
cosec ( x ) = − 2 1 sin x = − 2 sin x = − 1 2
The value,
sin 30 ° = 1 2
The value of x where sin is negative is,
In the third and fourth quadrant, sin is negative,
Value in the second quadrant is,
180 ° + 30 ° = 210 °
And the value in the fourth quadrant is,
360 ° − 30 ° = 330 °
So, the principal solutions are,
x = 210 ° x = 210 ° × π 180 ° x = 7 π 6
And,
x = 330 ° x = 330 ° × π 180 ° x = 11 π 6
To find the general solution,
Consider,
sin x = sin y (1)
sin x = − 1 2 (2)
From equations (1) and (2),
sin y = − 1 3 sin y = sin 7 π 6 y = 7 π 6
Since, sin x = sin y
General solution is,
x = 7 π 6
For x = 7 π 6
Thus, x = n π + ( − 1 ) n 7 π 6 where, n ∈ z
Given that,
The value,
The value of
In the third and fourth quadrant,
Value in the second quadrant is,
And the value in the fourth quadrant is,
So, the principal solutions are,
And,
To find the general solution,
Consider,
From equations (1) and (2),
Since,
General solution is,
For
Thus,
