Question

in which quadrant would \\( \theta \\) be if \\( \cos \theta=-\frac{1}{2} \\) and \\( \tan \theta<0 \\)? use a digit for your response rather than a roman numeral. (1 point) quadrant check answer remaining attempts : 3

Explanation:

Step1: Analyze cosine sign

Cosine is negative, so $\theta$ is in quadrants where $x$-coordinate is negative: Quadrants II or III (since $\cos\theta=\frac{x}{r}$, $r>0$, so $x<0$ implies Quadrants II, III).

Step2: Analyze tangent sign

Tangent is $\frac{\sin\theta}{\cos\theta}$. $\tan\theta<0$ means sine and cosine have opposite signs. Since $\cos\theta<0$, $\sin\theta$ must be positive (to make $\tan\theta=\frac{+}{-}<0$).

Step3: Determine quadrant

Sine positive (y - positive) and cosine negative (x - negative) occurs in Quadrant II. In digit form, Quadrant II is 2.

Answer:

2