Question

Find the missing trig functions.... csc θ=-4, cos θ < 0

Answer 1

First, theta is an angle that MUST be in quadrant III. Why? Because you are given that csc theta = -4 and cos theta < 0, so only in quadrant III will csc be negative and cosine be negative. That MUST be established before even starting this problem.

Now...we can start solving...
sin theta is the reciprocal function of csc theta, so if csc theta is - 4, then sin theta = - 1/4.
Now....to figure the remaining four trig functions..you need a diagram of the triangle in quadrant three, where csc = -4/1.
csc theta = length of the hypotenuse/length of opposite leg.
so the remaining leg must be - sqrt (15)

so now..you compute the remaining 4 trig functions.