Use a TI-84 calculator.

Since the interval is [0,2π) and not [0,360o), make sure the
calculator is in radian mode.  Press MODE, and make sure RADIAN is highlighted
and not DEGREE. If not, scroll to RADIAN and press ENTER. Press 2ND MODE to get
back to the main screen.

First we'll find each values in QI, then we'll find the value in the other
quadrant.

The calculator doesn't handle the reciprocal trig ratios, which are
cotangent, cosecant, and secant directly.  It only handles sine, 
cosine, and tangent directly. The others are found from them.

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a) cot(x) = 8

8 is positive and the cotangent is positive in QI and QIII.
We always find the QI value first.

We must first convert cotangent of 8 to a tangent 

Press 8 
Press x-1
Press ENTER
Read 0.125

Then take the inverse tangent of that:"

Press 2ND 
Press TAN
Type 0.125
Press )
Press ENTER
Read 0.1243549945   <--that's the QI value for a)

We want the QI value, so 0.1243549945 is one of the answers to a).

But we also want the QIII value, so we add to π

But we also want the QIII value, so we add π

Press + 2ND ^
Your screen should read Ans+π
Press ENTER
Read 3.265947648   <--that's the QIII answer.

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b) cscx=-3 

-3 is negative and the cosecant is negative in QIII and QIV.
We always find the QI value first.

We must first convert cosecant, which is -3 to a sine 

Press 3      <--NOTE: press 3, not -3 (3 is the absolute value of -3) 
Press x-1
Press ENTER
Read 0.3333333333

Then take the inverse sine of that:"

Press 2ND 
Press SIN
Type 0.3333333333
Press )
Press ENTER
Read 0.3398369094   <--that's the QI value, which we don't want.

We want the QIII value, so we add π

Press + 2ND ^
Your screen should read Ans+π
Press ENTER
Read 3.481429563   <--that's the QIII answer.

But we also want the QIV value, so we subtract from 2π

Press 2 2ND ^ - 0.3398369094 
You should read 
2π-0.3398369094
Press ENTER
Read 5.943348398   <--that's the QIV answer.

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c) sec(x) = 7

7 is positive and the secant is positive in QI and QIV.
We always find the QI value first.

We must first convert the secant, 8, to a cosine 

Press 7
Press x-1
Press ENTER
Read 0.1428571429

Then take the inverse cosine of that:"

Press 2ND 
Press COS
Type 0.1428571429
Press )
Press ENTER
Read 1.427448758   <--that's the QI value for c)

We want the QI value, so 1.427448758 is one of the answers to c).

But we also want the QIV value, so we subtract from 2π

Press 2 2ND ^ - 1.427448758 
You should read 
2π-1.47448758
Press ENTER
Read 4.855736549   <--that's the QIV answer.

Edwin