Question

Acute angle w
has sin w=4/97
and tan w =4/9
. Which are values of csc w
and cot w
?


and



and



and



and

Answer 1

They cannot both be true for the same acute angle (sin = 4/97 and tan = 4/9 are inconsistent). Treating them separately:

1) If sin w = 4/97 (acute):
- csc w = 1/sin w = 97/4.
- cot w = adjacent/opposite = sqrt(hyp^2 - opp^2)/opp = sqrt(97^2 − 4^2)/4 = sqrt(9393)/4.

2) If tan w = 4/9 (acute):
- cot w = 1/tan w = 9/4.
- sin w = opposite/hyp = 4/√97, so csc w = √97/4.