Let's start with the given equation:
[tex]\[ F \sin x = \frac{3}{5} \][/tex]
Knowing that [tex]\( F = 1 \)[/tex] in this context, we have:
[tex]\[ \sin x = \frac{3}{5} \][/tex]
Next, we need to find the value of [tex]\( \frac{\cos x + \tan x}{\sin x} \)[/tex]. Let's break this down into steps:
1. Find [tex]\(\cos x\)[/tex]:
We use the Pythagorean identity to find [tex]\(\cos x\)[/tex]:
[tex]\[ \sin^2 x + \cos^2 x = 1 \][/tex]
Substituting [tex]\(\sin x = \frac{3}{5}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^2 + \cos^2 x = 1 \][/tex]
[tex]\[ \frac{9}{25} + \cos^2 x = 1 \][/tex]
[tex]\[ \cos^2 x = 1 - \frac{9}{25} \][/tex]
[tex]\[ \cos^2 x = \frac{25}{25} - \frac{9}{25} \][/tex]
[tex]\[ \cos^2 x = \frac{16}{25} \][/tex]
Taking the positive root (as we're not specified otherwise):
[tex]\[ \cos x = \frac{4}{5} \][/tex]
2. Find [tex]\(\tan x\)[/tex]:
We know that:
[tex]\[ \tan x = \frac{\sin x}{\cos x} \][/tex]
Substituting the values we have:
[tex]\[ \tan x = \frac{\frac{3}{5}}{\frac{4}{5}} \][/tex]
[tex]\[ \tan x = \frac{3}{4} \][/tex]
3. Calculate the expression [tex]\(\frac{\cos x + \tan x}{\sin x}\)[/tex]:
Substitute [tex]\(\cos x\)[/tex] and [tex]\(\tan x\)[/tex] into the expression:
[tex]\[ \frac{\cos x + \tan x}{\sin x} = \frac{\frac{4}{5} + \frac{3}{4}}{\frac{3}{5}} \][/tex]
4. Simplify the numerator:
Find a common denominator for [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ \frac{4}{5} = \frac{16}{20} \][/tex]
[tex]\[ \frac{3}{4} = \frac{15}{20} \][/tex]
Add the fractions:
[tex]\[ \frac{16}{20} + \frac{15}{20} = \frac{31}{20} \][/tex]
5. Combine the fractions:
Substitute back into the expression:
[tex]\[ \frac{\frac{31}{20}}{\frac{3}{5}} \][/tex]
Division of fractions:
[tex]\[ \frac{31}{20} \div \frac{3}{5} = \frac{31}{20} \times \frac{5}{3} \][/tex]
Multiply:
[tex]\[ \frac{31 \times 5}{20 \times 3} = \frac{155}{60} \][/tex]
Simplify:
[tex]\[ \frac{155}{60} = \frac{31}{12} \approx 2.5833 \][/tex]
Thus, the value of [tex]\(\frac{\cos x + \tan x}{\sin x}\)[/tex] is approximately [tex]\(2.5833\)[/tex].
