Question

If the value of tan θ<0 and sin θ>0 , then the angle θ must lie in which quadrant?

Answer 1

Given:

`\tan \theta<0\text{ and }\sin \theta>0`
`\text{ we know that }\tan \theta=(\sin \theta)/(\cos \theta)\text{.}`
`\text{ Replace tan}\theta=(\sin \theta)/(\cos \theta)\text{ in tan}\theta<0\text{ as follows.}`
`(\sin \theta)/(\cos \theta)<0`
`\text{Multiply cos}\theta\text{ on both sides.}`

`(\sin\theta)/(\cos\theta)*\cos \theta<0*\cos \theta`
`\sin \theta<0`
`\text{But given that sin}\theta>0`

hence sine value should be equal to 0.

`\sin \theta=0`
`\theta=\sin ^(-1)0`
`\theta=\sin ^(-1)\sin (n\pi)`
`\theta=n\pi\text{ where n is integer.}`