Let's find out the correct options based on the given value [tex]\(\sec \theta=\frac{13}{12}\)[/tex].
1. Calculate [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
Given:
[tex]\[
\sec \theta = \frac{13}{12}
\][/tex]
Therefore:
[tex]\[
\cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{13}{12}} = \frac{12}{13}
\][/tex]
So, option A, [tex]\(\cos \theta = \frac{12}{13}\)[/tex], is TRUE.
2. Calculate [tex]\(\sin \theta\)[/tex]:
Using the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
With [tex]\(\cos \theta = \frac{12}{13}\)[/tex]:
[tex]\[
\left(\frac{12}{13}\right)^2 + \sin^2 \theta = 1
\][/tex]
[tex]\[
\frac{144}{169} + \sin^2 \theta = 1
\][/tex]
[tex]\[
\sin^2 \theta = 1 - \frac{144}{169} = \frac{169}{169} - \frac{144}{169} = \frac{25}{169}
\][/tex]
[tex]\[
\sin \theta = \sqrt{\frac{25}{169}} = \frac{5}{13}
\][/tex]
So, option B, [tex]\(\sin \theta = \frac{12}{13}\)[/tex], is FALSE.
3. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
With [tex]\(\sin \theta = \frac{5}{13}\)[/tex] and [tex]\(\cos \theta = \frac{12}{13}\)[/tex]:
[tex]\[
\tan \theta = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}
\][/tex]
So, option C, [tex]\(\tan \theta = \frac{5}{12}\)[/tex], is FALSE.
4. Calculate [tex]\(\csc \theta\)[/tex]:
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
With [tex]\(\sin \theta = \frac{5}{13}\)[/tex]:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5}
\][/tex]
So, option D, [tex]\(\csc \theta = \frac{12}{13}\)[/tex], is FALSE.
Therefore, the correct answer is:
- A: [tex]\(\cos \theta = \frac{12}{13}\)[/tex].
