Find the exact value of each of the remaining trigonometric functions of theta. tan theta = -1/5, csc theta > 0

Answer 1

If csc(θ) > 0 then sin(θ) > 0 and that occurs in quadrants 1 & 2. So we are looking for an angle in Q1 and/or Q2.

If tan(θ) is negative, then that further restricts the angle to Q2. Since tan(θ) = opposite/adjacent. then the opposite side must be 1 and the adjacent side must be -5 like this:

Using the Pythagorean Theorem, we can solve for h:

h = √(1² + (-5)²)

h = √26

So now we can write the exact value of the other trig function

sin(θ) = opp/hyp = 1/√26 = √26/26

cos(θ) = adj/hyp = -5/√26 = -5√26/26

csc(θ) = hyp/opp = √26/1 = √26

sec(θ) = hyp/adj = √26/-5

cot(θ) = adj/opp = -5/1 = -5