Let's solve the given problem step by step.
Given the equation [tex]\(5 \tan \beta = 4\)[/tex], we want to determine the value of [tex]\(\frac{5 \sin \beta - 2 \cos \beta}{5 \sin \beta + 2 \cos \beta}\)[/tex].
1. Express [tex]\(\tan \beta\)[/tex] in a simplified form:
[tex]\[
\tan \beta = \frac{4}{5}
\][/tex]
Since [tex]\(\tan \beta = \frac{\sin \beta}{\cos \beta}\)[/tex], we can write:
[tex]\[
\frac{\sin \beta}{\cos \beta} = \frac{4}{5}
\][/tex]
2. Express [tex]\(\sin \beta\)[/tex] in terms of [tex]\(\cos \beta\)[/tex]:
[tex]\[
\sin \beta = \frac{4}{5} \cos \beta
\][/tex]
3. Use the Pythagorean identity [tex]\(\sin^2 \beta + \cos^2 \beta = 1\)[/tex] to find [tex]\(\cos \beta\)[/tex]:
[tex]\[
\left( \frac{4}{5} \cos \beta \right)^2 + \cos^2 \beta = 1
\][/tex]
[tex]\[
\frac{16}{25} \cos^2 \beta + \cos^2 \beta = 1
\][/tex]
[tex]\[
\left( \frac{16}{25} + 1 \right) \cos^2 \beta = 1
\][/tex]
[tex]\[
\frac{41}{25} \cos^2 \beta = 1
\][/tex]
[tex]\[
\cos^2 \beta = \frac{25}{41}
\][/tex]
Therefore:
[tex]\[
\cos \beta = \sqrt{ \frac{25}{41} } = \frac{5}{\sqrt{41}}
\][/tex]
4. Find [tex]\(\sin \beta\)[/tex]:
[tex]\[
\sin \beta = \frac{4}{5} \cos \beta = \frac{4}{5} \cdot \frac{5}{\sqrt{41}} = \frac{4}{\sqrt{41}}
\][/tex]
5. Calculate the numerator [tex]\(5 \sin \beta - 2 \cos \beta\)[/tex]:
[tex]\[
5 \sin \beta - 2 \cos \beta = 5 \left( \frac{4}{\sqrt{41}} \right) - 2 \left( \frac{5}{\sqrt{41}} \right)
\][/tex]
[tex]\[
= \frac{20}{\sqrt{41}} - \frac{10}{\sqrt{41}}
\][/tex]
[tex]\[
= \frac{10}{\sqrt{41}}
\][/tex]
6. Calculate the denominator [tex]\(5 \sin \beta + 2 \cos \beta\)[/tex]:
[tex]\[
5 \sin \beta + 2 \cos \beta = 5 \left( \frac{4}{\sqrt{41}} \right) + 2 \left( \frac{5}{\sqrt{41}} \right)
\][/tex]
[tex]\[
= \frac{20}{\sqrt{41}} + \frac{10}{\sqrt{41}}
\][/tex]
[tex]\[
= \frac{30}{\sqrt{41}}
\][/tex]
7. Form the required expression:
[tex]\[
\frac{5 \sin \beta - 2 \cos \beta}{5 \sin \beta + 2 \cos \beta} = \frac{\frac{10}{\sqrt{41}}}{\frac{30}{\sqrt{41}}} = \frac{10}{30} = \frac{1}{3}
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
