In I, all trig functions are positive
In II, the sine is positive.
In III, the tangent is positive. But tan(x)=sin(x)/cos(x), therefore both sine and cosine must be negative.
In IV, cosine is positive.
Are you telling me that you do NOT KNOW that tan(x) = 1/cot(x)? Does the reciprocal change the sign?
There is a reason why calculators ALWAYS have been built with only THREE trig functions. The user is expected to know some basic information and relationships concerning the other three.
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