Can someone explain why the angle has to be obtuse?

Question

If $\alpha$ is an angle in a triangle, and $\tan\alpha \leq 0$,

(a) $\alpha$ is an acute angle; (b) $\alpha$ is a right angle; (c) $\alpha$ is obtuse; (d) can't be determined.

Answer is c. No explanation provided.

Using the unit circle I can see how it could be obtuse. But I don't understand why it cant also be an acute angle.

Answer 1

It is given that $\alpha$ is an angle of a triangle. So it can either be acute, right or obtuse. But it is also given that $\tan\alpha\le0$ So this means that $\alpha$ can't be acute. Hence $\alpha$ can either be right or obtuse. But for $\tan\alpha=0$ $\alpha$ has to be equal to $0$ but this can't happen in a triangle so $\alpha$ has to be obtuse. This is the correct answer.

Now if you an explanation, then recall that $\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$ Now if $\alpha$ is acute or $\alpha\in(0,\pi/2)$ then both $\sin$ and $\cos$ are positive, hence $\tan\alpha$ is also positive. But since it is given that $\tan\alpha\le0$ it means that $\alpha$ can't be acute.

Answer 2

We know that $\tan x$ is negative in the second and fourth quadrant i.e. for $x \in (\pi/2, \pi)\ \text{or } x\in(3\pi/4, 2\pi)$.

Triangle can't have an angle $\geq 180^{\circ}$ so $\alpha$ should lie in 2$^{\text{nd}}$ quadrant i.e. $\alpha$ is obtuse.