To find [tex]\(\sec \theta\)[/tex], given the equation [tex]\(12 \cot \theta = 15\)[/tex], we can follow these steps:
1. Express [tex]\(\cot \theta\)[/tex] in simpler terms.
[tex]\[
\cot \theta = \frac{15}{12}
\][/tex]
Simplify this fraction:
[tex]\[
\cot \theta = \frac{5}{4}
\][/tex]
2. Find [tex]\(\tan \theta\)[/tex].
Since [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex],
[tex]\[
\tan \theta = \frac{1}{\cot \theta} = \frac{1}{\frac{5}{4}} = \frac{4}{5} = 0.8
\][/tex]
3. Use the Pythagorean identity to find [tex]\(\sec \theta\)[/tex].
We know from trigonometric identities that:
[tex]\[
\tan^2 \theta + 1 = \sec^2 \theta
\][/tex]
Substitute [tex]\(\tan \theta\)[/tex] with [tex]\(0.8\)[/tex]:
[tex]\[
(0.8)^2 + 1 = \sec^2 \theta
\][/tex]
Calculate [tex]\((0.8)^2\)[/tex]:
[tex]\[
0.64 + 1 = \sec^2 \theta
\][/tex]
[tex]\[
\sec^2 \theta = 1.64
\][/tex]
4. Find [tex]\(\sec \theta\)[/tex] by taking the square root.
[tex]\[
\sec \theta = \sqrt{1.64}
\][/tex]
Calculate the square root:
[tex]\[
\sec \theta = 1.2806248474865698
\][/tex]
So, [tex]\(\sec \theta = 1.2806248474865698\)[/tex].
