Math~1

a) sin theta = -0.7833

=> theta = sin-1 (-0.7833)

= -51.564 (degree)

= - (51.564/180)*pi (radian)

= -0.286 * pi (radian)

-ve value of sin theta means, 3rd Quarter and 4th Quarter by ASTC rule.
moreover, cos theta >0 mean +ve value of cos theta, 1st and 4th Quarter.

to satify both condition, only 4th Quarter is correct,

therefore, the value of theta is (2pi - 0.286*pi) =>1.714*pi.

the rest of the questions is similiar method of Question "a", but you need to clear 2 things. one is the ASTC rule and the answer request in radian, not degree, need to change the answer in (pi).

P.S

ASTC rule

A -> All +ve, mean Sin, Cos and Tan > 0
S -> Sin +ve, mean Sin >0 but (Cos and Tan <0)
T -> Tan +ve, mean Tan >0 but (Cos and Sin <0)
C -> Cos +ve, mean Cos >0 but (Sin and Tan <0)

you just use the table of signs of trigonometric functions
For example
a) sin theta = -0.7833, cos theta > 0
consider
a) sin x = 0.7833
x=51.56
Since if sin theta <0, cos theta > 0
Then theta should be at the fourth quadrant
So the required theta
=360-x
=360-51.56
=308.44
(b)
cos x=0.5898
x=53.86
theta
=53.86
(c)
cos x=0.5646
x=55.63
theta
=124.37
(d)
tanx=2.572
x=68.75
theta
=180+68.75
=248.75