Let's analyze the given information and go through the solution step-by-step to determine if each statement is true or false.
Given:
[tex]\[ \csc \theta = \frac{13}{5} \][/tex]
We know that [tex]\(\csc \theta\)[/tex] is the reciprocal of [tex]\(\sin \theta\)[/tex]. Therefore,
[tex]\[ \sin \theta = \frac{1}{\csc \theta} = \frac{5}{13} \][/tex]
Next, we use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substituting [tex]\(\sin \theta = \frac{5}{13}\)[/tex],
[tex]\[ \left( \frac{5}{13} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{25}{169} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{25}{169} \][/tex]
[tex]\[ \cos^2 \theta = \frac{169}{169} - \frac{25}{169} \][/tex]
[tex]\[ \cos^2 \theta = \frac{144}{169} \][/tex]
[tex]\[ \cos \theta = \sqrt{\frac{144}{169}} \][/tex]
[tex]\[ \cos \theta = \frac{12}{13} \][/tex]
Now that we have [tex]\(\cos \theta\)[/tex], we can find [tex]\(\sec \theta\)[/tex], [tex]\(\tan \theta\)[/tex], [tex]\(\sin \theta\)[/tex], and [tex]\(\cos \theta\)[/tex]:
1. [tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex]:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12} \][/tex]
2. [tex]\(\tan \theta\)[/tex] is given by:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12} \][/tex]
3. We already have [tex]\(\sin \theta\)[/tex]:
[tex]\[ \sin \theta = \frac{5}{13} \][/tex]
4. We already have [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \frac{12}{13} \][/tex]
Now let's validate each of the given conditions:
A. [tex]\(\sec \theta = \frac{5}{13}\)[/tex]
- We found [tex]\(\sec \theta = \frac{13}{12}\)[/tex].
- Therefore, this statement is false.
B. [tex]\(\tan \theta = \frac{5}{12}\)[/tex]
- We found [tex]\(\tan \theta = \frac{5}{12}\)[/tex].
- Therefore, this statement is true.
C. [tex]\(\sin \theta = \frac{5}{13}\)[/tex]
- We found [tex]\(\sin \theta = \frac{5}{13}\)[/tex].
- Therefore, this statement is true.
D. [tex]\(\cos \theta = \frac{5}{13}\)[/tex]
- We found [tex]\(\cos \theta = \frac{12}{13}\)[/tex].
- Therefore, this statement is false.
So the truth values are:
- A: False
- B: True
- C: True
- D: False
