Answer :

GinnyAnswer
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].

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