Question

If sin theta <0 and tan theta >0 then:

Answer 1

Answer:

Step-by-step explanation:

If the sine is negative then theta must be in Quadrant III or Quadrant IV.

If the tangent is positive then theta must be in Quadrant I or III.

If both conditions must be satisfied, then we conclude that thera is in Quadrant III.

Answer 2

Note that $\tan\theta=\frac{\sin\theta}{\cos\theta}.$ Thus, if $\sin\theta<0$ and $\tan\theta>0,$ then we must have $\cos\theta<0$ as well. The sine function is negative when $\theta$ lies in Quadrant III or IV, and the cosine function is negative when $\theta$ lies in Quadrant II or III. Thus, the two are both negative when $\theta$ lies in Quadrant III, or $\pi+2\pi n<\theta<\frac{3\pi}{2}+2\pi n$ for integer values of $n.$