Question

determine the quadrant in which the terminal side of θ lies. (a) cosθ < 0 and sinθ > 0 (choose one) (b) sinθ < 0 and tanθ < 0 (choose one)

Explanation:

Step1: Recall trig - function signs in quadrants

In the coordinate plane, for an angle $\theta$: $\cos\theta$ is the $x$ - coordinate of the point on the unit - circle corresponding to $\theta$, $\sin\theta$ is the $y$ - coordinate, and $\tan\theta=\frac{\sin\theta}{\cos\theta}$.

Step2: Analyze part (a)

If $\cos\theta<0$, the $x$ - coordinate of the point on the unit - circle is negative. If $\sin\theta > 0$, the $y$ - coordinate of the point on the unit - circle is positive. In the four quadrants, this occurs in the second quadrant where $x<0$ and $y > 0$.

Step3: Analyze part (b)

If $\sin\theta<0$, the $y$ - coordinate of the point on the unit - circle is negative. If $\tan\theta=\frac{\sin\theta}{\cos\theta}<0$, and $\sin\theta<0$, then $\cos\theta>0$ (since a negative divided by a positive is negative). In the four quadrants, this occurs in the fourth quadrant where $x>0$ and $y < 0$.

Answer:

(a) Second quadrant
(b) Fourth quadrant