Exact value of csc theta and tan theta

Question

Let data be an angle in quadrant III such as cos theta equals negative for over five. Find the exact values of cosecant theta and tangent theta

Answer 1

It's a 3-4-5 right triangle in quadrant III, where both cosines, secants, cosecants and sines are negative, but tangents and cotangents are positive

sintheta = -3/5 4 is the adjacent side, 3 the opposite side, 5 the hypotenuse; sintheta = opposite/hypotenuse

csctheta = 1/sintheta = -5/3

tantheta opposite/adjacent = 3/4

Answer 2

cosς = -4/5 and ς in quadrant III. So, sinξ = - √(1 - (- 4/5)²) = - 3/5.

Now, cscξ = 1/sinς = -5/3 and tanς = sinς/cosς =(- 3/5)/(- 4/5) = 3/4