express each of the following as the same function
of a positive acute angle:
a. cos( 112°33')

That angle is in the second quadrant.
The cosine is negative in the second quadrant.
The reference angle is found by subtracting it from 180°

 180°       179°60'
-112°30' = -112°33'
-------------------
             67°27'

Therefore cos(112°33) = -cos(67°27') 

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b.tan(310°)
That angle is in the fourth quadrant.
The tangent is negative in the fourth quadrant.
The reference angle is found by subtracting 310° from 360°

 360°              
-310                  
-----                  
  50°

Therefore tan(310°) = -tan(50°)

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c.cot(138°15'10")

That angle is in the second quadrant.
The cotangent is negative in the second quadrant.
The reference angle is found by subtracting it from 180°

 180°          179°59'60"
-138°15'10" = -138°15'10"
-------------------------
                41°44'50"

Therefore cot(138°15'10") = -cot(41°44'50) 

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d.sin (-140°)

That angle is in the third quadrant.
The sine is negative in the third quadrant.
The reference angle is found by subtracting 140° from 180°

 180°              
-140°                  
-----                  
  40°

Therefore sin(-140°) = -sin(40°)

Edwin