Given that [tex]\(\sin \theta = \frac{12}{13}\)[/tex], we need to determine the value of the expression [tex]\(\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta}\)[/tex].
Here are the steps to solving this:
1. Determine [tex]\(\cos \theta\)[/tex]:
Since [tex]\(\sin \theta = \frac{12}{13}\)[/tex], we can use the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substituting the known value of [tex]\(\sin \theta\)[/tex] into the identity:
[tex]\[
\left(\frac{12}{13}\right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\frac{144}{169} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{144}{169}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{25}{169}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{25}{169}}
\][/tex]
[tex]\[
\cos \theta = \frac{5}{13}
\][/tex]
Since [tex]\(\theta\)[/tex] is in a range where [tex]\(\sin \theta\)[/tex] is positive and [tex]\(\cos \theta\)[/tex] is also positive, we have [tex]\(\cos \theta = \frac{5}{13}\)[/tex].
2. Calculate [tex]\(\sec \theta\)[/tex]:
[tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5} = 2.6
\][/tex]
3. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\(\tan \theta\)[/tex] is the ratio of [tex]\(\sin \theta\)[/tex] to [tex]\(\cos \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} = 2.4
\][/tex]
4. Calculate the expression [tex]\(\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta}\)[/tex]:
Now, substitute the values of [tex]\(\sec \theta\)[/tex] and [tex]\(\tan \theta\)[/tex]:
[tex]\[
\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2.6 + 2.4}{2.6 - 2.4} = \frac{5.0}{0.2} = 25.0
\][/tex]
Therefore, the value of the expression [tex]\(\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta}\)[/tex] is 25.
