To solve the problem, we start with the given equation [tex]\( 5 \cot \theta = 7 \)[/tex].
Step-by-Step Solution:
1. Solve for [tex]\( \cot \theta \)[/tex]:
[tex]\[
\cot \theta = \frac{7}{5}
\][/tex]
2. Express [tex]\( \cot \theta \)[/tex] in terms of sine and cosine:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
So, we have:
[tex]\[
\frac{\cos \theta}{\sin \theta} = \frac{7}{5}
\][/tex]
3. Solve for [tex]\( \tan \theta \)[/tex]:
[tex]\[
\tan \theta = \frac{1}{\cot \theta} = \frac{5}{7}
\][/tex]
4. Determine [tex]\( \sin \theta \)[/tex] and [tex]\( \cos \theta \)[/tex] using trigonometric identities:
To find expressions for [tex]\( \sin \theta \)[/tex] and [tex]\( \cos \theta \)[/tex], we use:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Given [tex]\( \tan \theta = \frac{5}{7} \)[/tex]:
[tex]\[
\sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} = \frac{\frac{5}{7}}{\sqrt{1 + \left(\frac{5}{7}\right)^2}} = \frac{5 / 7}{\sqrt{1 + 25 / 49}} = \frac{5 / 7}{\sqrt{\frac{74}{49}}} = \frac{5/7}{\sqrt{74}/7} = \frac{5}{\sqrt{74}}
\][/tex]
Similarly, for [tex]\( \cos \theta \)[/tex]:
[tex]\[
\cos \theta = \frac{1}{\sqrt{1 + \tan^2 \theta}} = \frac{1}{\sqrt{1 + \frac{25}{49}}} = \frac{1}{\sqrt{\frac{74}{49}}} = \frac{1}{\sqrt{74 / 49}} = \frac{7}{\sqrt{74}}
\][/tex]
5. Calculate [tex]\( 7 \sin \theta \)[/tex] and [tex]\( 5 \cos \theta \)[/tex]:
[tex]\[
7 \sin \theta = 7 \cdot \frac{5}{\sqrt{74}} = \frac{35}{\sqrt{74}}
\][/tex]
[tex]\[
5 \cos \theta = 5 \cdot \frac{7}{\sqrt{74}} = \frac{35}{\sqrt{74}}
\][/tex]
6. Form and simplify the numerator:
[tex]\[
7 \sin \theta + 5 \cos \theta = \frac{35}{\sqrt{74}} + \frac{35}{\sqrt{74}} = \frac{70}{\sqrt{74}}
\][/tex]
7. Calculate [tex]\( 5 \sin \theta \)[/tex] and [tex]\( 7 \cos \theta \)[/tex]:
[tex]\[
5 \sin \theta = 5 \cdot \frac{5}{\sqrt{74}} = \frac{25}{\sqrt{74}}
\][/tex]
[tex]\[
7 \cos \theta = 7 \cdot \frac{7}{\sqrt{74}} = \frac{49}{\sqrt{74}}
\][/tex]
8. Form and simplify the denominator:
[tex]\[
5 \sin \theta + 7 \cos \theta = \frac{25}{\sqrt{74}} + \frac{49}{\sqrt{74}} = \frac{74}{\sqrt{74}} = \sqrt{74} = 1
\][/tex]
9. Find the final value:
[tex]\[
\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta} = \frac{\frac{70}{\sqrt{74}}}{\sqrt{74}} = \frac{70}{74} = 0.945945945945946
\][/tex]
Hence, the value of [tex]\( \frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta} \)[/tex] is approximately [tex]\( 0.945945945945946 \)[/tex].
