Problems that i am stumped on is the sec of -4pi/3, and the sin of -23pi/6. the would be appreciated.

Here, As we are dealing with negative radian, we will go clockwise instead of counter-clock wise which we usually do in positive cases, 

So, Terminal values are, 0, -π/2 , -π, -3π/2, 

Now, You have: Sec (-4π/3)

Here, We can figure out, -π/3 is in between -π and -3π/2

so reference value = -π - (-4π/3) = Sec (π/3)
We know, Sec (π/3) = 2
As quadrant is 2th which is negative for sec, it would be -2

In short, Your Final Answer would be -2

Hope this helps!