If csc theta = -5/4 and 180 < theta < 270 degrees, find tan theta

a. -4/3
b. 3/4
c. 4/3
d. -4/5

Hello Cassandra L, cot^2 @ = csc^2 @ - 1
===> (-5/4)^2 - 1 = 25/16 - 1 = 9/16
===> tan^2 @ = 16/9
====> tan @ = 4/3 or -4/3
Given @ lies within 180 and 270 so tan @ has to be positive (third quadrant)
So tan @ = 4/3
Hence option C is the right one.

cscx = -5/4 and we know that 1 + (cotx)(cotx) = (cscx)(cscx) so cotx = square root ((cscx)(cscx)-1) = square root (25/16 -1) = 3/4 (PS: + 3/4 not -3/4 because theta is between 180 and 270.)

so tan x = 1/cotx = 4/3 (here x is theta)
hope i helped u
cheers

1 / sin Ө = - 5/4

sin Ө = - 4/5

tan Ө = - 4/3 <-------Option a