What are the solutions to csc x + 2 = 0 for 0 <= x < 2 \pi?

[iBankingFTW]

[SOLVED] Solving csc x + 2 for 0

Homework Statement

"Solve csc x + 2 = 0 for 0 <= x < 2 \pi

Choices are:

A. \pi/6 and 5\pi/6
B. \pi/6 and 7\pi/6
C. 4\pi/3 and 5\pi/3
D. 7\pi/6 and 11\pi/6

Homework Equations

csc x =2
sin x = 1/2 = 30 degrees = \pi/6

The Attempt at a Solution

From csc x + 2 = 0

I get csc x = 2

Which is:

1/sin = 1/2 and I know that 1/2 is \pi/6


So the answer is either A or B but I don't understand where the second answer comes from.

 

[rocomath]

You messed up your Algebra. You will get 2 solutions b/c Cosecant takes on that value at 2 places. Sine is negative in what Quadrants? Thus, Cosecant is also negative at those 2 places.

 

[iBankingFTW]

Ohhhh, ok. So \pi/6 is in Quadrant I so that's correct and then 5\pi/6 is in Quad. 2 so that's also correct but 7\pi/6 is in Quad. 3 which is negative.

So the answer is A.?

 

[rocomath]

No. Where is Sine negative? Definitely not Quad 2 & 3.

 

[iBankingFTW]

Tangent is positive in Quadrant 3 (ASTC as I learned it), so wouldn't sine be negative there?

I asked my dad about this too and he said that the answer is B. but I don't understand because 7\pi/6 is in the third quadrant and isn't that negative if it's sine?

 

[rocomath]

Did you fix your first step?

You're not solving for \csc x=2 ... it's \csc x=-2

Check your Algebra again! So your values should be in Quadrants 3 & 4 ...

 

[iBankingFTW]

Oh jeez, I'm such an idiot. I hate making little mistakes like that. So the answer is B. then since 7\pi/6 is in Quadrant 3?

 

[rocomath]

Oh jeez, I'm such an idiot. I hate making little mistakes like that. So the answer is B. then since 7\pi/6 is in Quadrant 3?

Where else? One more solution!

 

[iBankingFTW]

Haha...thanks for the help.