Question

determine whether the following statement makes sense or does not make sense, and explain your reasoning. this angle θ is in a quadrant in which cot θ>0 and tan θ<0. choose the correct answer below. a. the statement makes sense because there is at least one angle in the first quadrant for which cot θ>0 and tan θ<0. b. the statement does not make sense because there is no angle θ, and no quadrant in which cot θ>0 and tan θ<0. c. the statement makes sense because there is at least one angle in the third quadrant for which cot θ>0 and tan θ<0. d. the statement makes sense because there is at least one angle in the fourth quadrant for which cot θ>0 and tan θ<0. e. the statement makes sense because there is at least one angle in the second quadrant for which cot θ>0 and tan θ<0.

Explanation:

Step1: Recall the relationship between cotangent and tangent

Recall that $\cot\theta=\frac{1}{\tan\theta}$. So, $\cot\theta$ and $\tan\theta$ always have the same sign.

Step2: Analyze the given condition

The condition is $\cot\theta > 0$ and $\tan\theta<0$. Since $\cot\theta$ and $\tan\theta$ have the same - sign, it is impossible for $\cot\theta$ to be positive and $\tan\theta$ to be negative.

Answer:

B. The statement does not make sense because there is no angle $\theta$, and no quadrant in which $\cot\theta > 0$ and $\tan\theta<0$.