To solve the given problem, we need to find the value of [tex]\( \frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 3 \cos \theta} \)[/tex]. Given that [tex]\( 5 \cos \theta = 3 \)[/tex], let's go through the steps to determine the required value.
1. Find [tex]\(\cos \theta\)[/tex]:
[tex]\[
5 \cos \theta = 3 \implies \cos \theta = \frac{3}{5}
\][/tex]
2. Determine [tex]\(\sin \theta\)[/tex]:
Utilize the trigonometric identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[
\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1
\][/tex]
[tex]\[
\sin^2 \theta + \frac{9}{25} = 1
\][/tex]
[tex]\[
\sin^2 \theta = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\sin^2 \theta = \frac{25}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\sin^2 \theta = \frac{16}{25}
\][/tex]
[tex]\[
\sin \theta = \sqrt{\frac{16}{25}}
\][/tex]
[tex]\[
\sin \theta = \frac{4}{5}
\][/tex]
3. Evaluate the numerator [tex]\(5 \sin \theta - 3 \cos \theta\)[/tex]:
[tex]\[
5 \sin \theta - 3 \cos \theta = 5 \cdot \frac{4}{5} - 3 \cdot \frac{3}{5}
\][/tex]
[tex]\[
= 4 - \frac{9}{5}
\][/tex]
[tex]\[
= 4 - 1.8
\][/tex]
[tex]\[
= 2.2
\][/tex]
4. Evaluate the denominator [tex]\(5 \sin \theta + 3 \cos \theta\)[/tex]:
[tex]\[
5 \sin \theta + 3 \cos \theta = 5 \cdot \frac{4}{5} + 3 \cdot \frac{3}{5}
\][/tex]
[tex]\[
= 4 + \frac{9}{5}
\][/tex]
[tex]\[
= 4 + 1.8
\][/tex]
[tex]\[
= 5.8
\][/tex]
5. Calculate the required value:
[tex]\[
\frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 3 \cos \theta} = \frac{2.2}{5.8}
\][/tex]
[tex]\[
\approx 0.37931034482758624
\][/tex]
Therefore, the required value is approximately [tex]\( 0.3793 \)[/tex].
