Hello Mike,
First, we can find a co-terminal angle of -11pi/3. It can be done by adding or subtracting 2π as many as needed, but here, we'll be adding until we get a positive angle that shows up on your unit circle.
So: -11pi/3 + 2pi = -11pi/3 + 6pi/3 = -5pi/3
----> Let's add another 2pi: -5pi/3 + 2pi = -5pi/3 + 6pi/3 = pi/3
We have a positive angle whose sine and cosine values we know from the unit circle. Nice!
We can turn this problem from "finding csc(-11pi/3)" to "finding csc(pi/3)" because -11pi/3 and pi/3 are co-terminal angles, and co-terminal angles share the same sin,cos,tan, etc.
Your unit circle does not explicitly tell you what csc(pi/3) equals, but it does tell you sin(pi/3) = √3/2
Also: csc is the reciprocal of sin: csc x = 1/sinx
Therefore: csc(pi/3) = 1/sin(pi/3) = 1/(√3/2) = 2/√3
Rationalize the denominator by multiplying the previous answer by √3/√3 and you get: csc(pi/3) = 2√3/3
Remember that -11pi/3 and pi/3 are co-terminal angles, and co-terminal angles share the same sin,cos,tan, etc. Therefore: csc(-11pi/3) also equals 2√3/3
Cheers.
