find the exact value: csec 22pi/3

Is this csec as in cosecant? If so, that would be csc 22pi/3. Through trigonometric identities, csc is just the inverse of sin. So it is equivalent to 1/sin(22pi/3). Make sure that your calculator is in radian mode. Then input this:

1/sin(22pi/3) = -1.15

If you actually meant sec 22pi/3, then that is equivalent to 1/cos(22pi/3):

1/cos(22pi/3) = -2

Either of those two is the answer :)