Question

Given `tan theta < 0`, `sin theta < 0` and `sec theta < 0`, `tan theta < 0`, which quadrants does both angles theta lie on?

Answer 1

Both `tan theta and sin theta` are < 0 in the fourth quadrant Q4. Both `sec theta and tan theta` are < 0 in Q2..

Explanation:

All are > 0 for `theta` in the first quadrant `Q1(0, pi/2)`,

sine and csc only are > 0 for `theta in Q2(pi/2, pi)`,

tangent and cotangent only are > 0 for `theta in Q3(pi. (3pi)/2)` and

cosine and secant only are > 0 for `theta in Q4((3pi)/2, pi)`.

Accordingly, both `tan theta and sin theta` are < 0 in the fourth quadrant Q4.

Both `sec theta and tan theta` are < 0 in in Q2..

I have used open interval notation (..) for the intervals, owing to discontinuity-limit problems like the limits `tan (( pi/2)_(+-))=-+oo`, as `theta to pi/2`, from Q2 and Q1, respectively...