Given [tex]\(\cot(\beta) = \frac{\sqrt{3}}{7}\)[/tex], we want to find [tex]\(\sec(\beta)\)[/tex].
### Step-by-Step Solution:
1. Find [tex]\(\tan(\beta)\)[/tex]:
Given the cotangent of [tex]\(\beta\)[/tex],
[tex]\[ \cot(\beta) = \frac{\sqrt{3}}{7} \][/tex]
we know the relationship between cotangent and tangent:
[tex]\[ \tan(\beta) = \frac{1}{\cot(\beta)} \][/tex]
Substituting the given value,
[tex]\[ \tan(\beta) = \frac{1}{\frac{\sqrt{3}}{7}} = \frac{7}{\sqrt{3}} \][/tex]
2. Simplify [tex]\(\tan(\beta)\)[/tex]:
[tex]\[ \tan(\beta) = \frac{7\sqrt{3}}{3} \approx 4.04145 \][/tex]
3. Find [tex]\(\cos(\beta)\)[/tex]:
We use the identity involving tangent and secant:
[tex]\[ \sec^2(\beta) = 1 + \tan^2(\beta) \][/tex]
First, compute [tex]\(\tan^2(\beta)\)[/tex]:
[tex]\[ \tan^2(\beta) = \left( \frac{7\sqrt{3}}{3} \right)^2 = \frac{49 \cdot 3}{9} = \frac{147}{9} = 16.3333 \][/tex]
Then, use this value to find [tex]\(\sec^2(\beta)\)[/tex]:
[tex]\[ \sec^2(\beta) = 1 + \tan^2(\beta) = 1 + 16.3333 = 17.3333 \][/tex]
4. Find [tex]\(\sec(\beta)\)[/tex]:
Take the square root of both sides to find [tex]\(\sec(\beta)\)[/tex]:
[tex]\[ \sec(\beta) = \sqrt{17.3333} \approx 4.1633 \][/tex]
Therefore, [tex]\(\sec(\beta) \approx 4.1633 \)[/tex].
This is an approximate value, rounding to four significant figures if necessary.
