Question

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

Answer 1

Note that
`\tan\theta=(\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<(3\pi)/(2)+2\pi n` for integer values of
`n.`

Answer 2

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.