Solve for x with a trigonometric function

[ver_mathstats]

Homework Statement

Solve for x.

Relevant Equations

sin(3x)= -1/2

Homework Statement: Solve for x.
Homework Equations: sin(3x)= -1/2

sin(3x) = -1/2

3x = sin^-1(-1/2)

3x = -π/6

x = -π/18

x = -π/18 + 2π/3 = 11π/18

11π/18 + 2π/3 = 23π/18

11π/18 + (2π(4))/3 = 35π/18

The solutions I obtained were 23π/18 and 35π/18. Are these correct? I'm not entirely sure if I did this problem correctly. Thank you.

 

[Charles Link]

What about a solution or solutions where the angle 3x is in the 3rd quadrant? (You found the solution when 3x is in the 4th quadrant). $\\$ Your solutions are incomplete, even for the 4th quadrant. You need to let $3x=\theta+n (2 \pi)$, where $n$ is any integer. You found $\theta$ for the 4th quadrant, etc., but it helps to be more systematic. Otherwise, you don't get the complete set.

 

[ver_mathstats]

What about a solution or solutions where the angle 3x is in the 3rd quadrant? (You found the solution when 3x is in the 4th quadrant). $\\$ Your solutions are incomplete, even for the 4th quadrant. You need to let $3x=\theta+n (2 \pi)$, where $n$ is any integer. You found $\theta$ for the 4th quadrant, etc., but it helps to be more systematic. Otherwise, you don't get the complete set.

Yes I can see where I went wrong. I got the solutions 7π/18, 11π/18, 19π/18, 23π/18, 31π/18, 35π/18 after looking it over. Thank you.

 

[Charles Link]

Still incomplete. And please show your work. And I don't think I agree with the $\frac{7 \pi}{18}$. Edit: My mistake. OP has it correct. As @PAllen mentions below though, the OP needs to include the negative x values that come from negative integer $n$ in $3x=\theta+n(2 \pi)$.

 

[PAllen]

And I don't think I agree with the $\frac{7 \pi}{18}$.

Looks fine to me. Also, his last set of solutions looks fine as the beginning of an obvious pattern that is complete (except for negative x values).

 

[Charles Link]

Looks fine to me. Also, his last set of solutions looks fine as the beginning of an obvious pattern that is complete (except for negative x values).

Sorry. My mistake. See the Edit above.