How do I find the other 5 trigonometric functions with just a cosecant value and what quadrant it is in?

Question

I know how to find sin (it is the reciprocal of csc) but with out a cos value I cannot figure out how to find tan, cot, cos and sec

Answer 1

let csc = 17/8 = hypotenuse/opposite side

sin = opposite side/hypotenuse = 8/17

adjacent side^2 + opposite side^2 = hypotenuse^2

adjacent side^2 + 8^2 = 17^2

adjacent side^2 + 64 = 289

adjacent side^2 = 225

Take the square root of both sides

adjacent side = 15

cos = adj/hyp = 15/17

sec = hyp/adj = 17/15

tan = opp/adj = 8/15

cot = adj/opp = 15/8

Answer 2

If you have sin Φ = x/hyp, couldn't you use this fact and a²+b² = c² to solve for the missing side of the triangle made by the angle? Once you know the length of all three sides you can write all the other trig functions.

The quadrant depends on the actual angle in question.