Given the statement, we need to determine which option cannot be true based on the provided values for [tex]\(\tan \theta = -\frac{12}{5}\)[/tex] and [tex]\(\csc \theta = -\frac{13}{12}\)[/tex].
Let's break it down step-by-step:
### Option A
"A. cannot be true because [tex]\(12^2 + 5^2 \neq 1\)[/tex]."
Calculating the sum of the squares:
[tex]\[ 12^2 + 5^2 = 144 + 25 = 169 \][/tex]
This is indeed the case; [tex]\(12^2 + 5^2 = 169\)[/tex] and not 1. However, this statement seems to check the Pythagorean theorem rather than the unit circle properties. So, let's move on to check the other options.
### Option B
"B. cannot be true because [tex]\(\tan \theta\)[/tex] must be less than 1."
We know:
[tex]\[ \tan \theta = -\frac{12}{5} \][/tex]
Calculating the value:
[tex]\[ -\frac{12}{5} = -2.4 \][/tex]
So, indeed, [tex]\(\tan \theta\)[/tex] here is less than 1.
### Option C
"C. cannot be true because if [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], then [tex]\(\csc \theta = \pm \frac{13}{5}\)[/tex]."
Given:
[tex]\[ \csc \theta = -\frac{13}{12} \][/tex]
To verify:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
Using the given triangle properties:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \Rightarrow \sin \theta = \frac{\tan \theta \cos \theta}{1} \][/tex]
[tex]\[ \sin \theta = -\frac{12}{13} \][/tex]
And therefore:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} = -\frac{13}{12} \][/tex]
Since [tex]\(\csc \theta = -\frac{13}{12}\)[/tex] matches the given value.
### Option D
"D. cannot be true because [tex]\(\tan \theta\)[/tex] is greater than zero in quadrant 3."
In Quadrant 3, both sine and cosine are negative, resulting in a positive tangent ([tex]\(\tan \theta = \frac{\sin \theta}{ \cos \theta} = (+)\)[/tex]).
Given:
[tex]\[ \tan \theta = -\frac{12}{5} \][/tex]
Since [tex]\(\tan \theta\)[/tex] is negative in this statement, it cannot be true that [tex]\(\tan \theta\)[/tex] is greater than zero in quadrant 3.
Thus, this confirms that:
[tex]\[ B. cannot be true because \(\tan \theta\) must be less than 1 .\][/tex]
Therefore, the final answer is:
[tex]\[ \boxed{B} \][/tex]
