Answer :

Certainly! Let's solve the problem step by step.

Given:
[tex]\[ \cos x = \frac{5}{13} \][/tex]

### Step 1: Find [tex]\(\sin x\)[/tex]

We will use the Pythagorean identity:
[tex]\[ \sin^2 x + \cos^2 x = 1 \][/tex]

Substitute [tex]\(\cos x = \frac{5}{13}\)[/tex] into the identity:
[tex]\[ \sin^2 x + \left(\frac{5}{13}\right)^2 = 1 \][/tex]

First, calculate [tex]\(\left(\frac{5}{13}\right)^2\)[/tex]:
[tex]\[ \left(\frac{5}{13}\right)^2 = \frac{25}{169} \][/tex]

Now, substitute this value back into the Pythagorean identity:
[tex]\[ \sin^2 x + \frac{25}{169} = 1 \][/tex]

Next, solve for [tex]\(\sin^2 x\)[/tex]:
[tex]\[ \sin^2 x = 1 - \frac{25}{169} \][/tex]

Convert 1 to a fraction with the same denominator:
[tex]\[ 1 = \frac{169}{169} \][/tex]

Then, subtract the fractions:
[tex]\[ \sin^2 x = \frac{169}{169} - \frac{25}{169} = \frac{144}{169} \][/tex]

Take the square root of both sides to find [tex]\(\sin x\)[/tex]:
[tex]\[ \sin x = \sqrt{\frac{144}{169}} \][/tex]

Simplify the square root:
[tex]\[ \sin x = \frac{\sqrt{144}}{\sqrt{169}} = \frac{12}{13} \][/tex]

Since the goal is to compute the principal value, we take the positive root:
[tex]\[ \sin x = \frac{12}{13} \][/tex]

### Step 2: Find [tex]\(\tan x\)[/tex]

We know that:
[tex]\[ \tan x = \frac{\sin x}{\cos x} \][/tex]

Substitute the values of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
[tex]\[ \tan x = \frac{\frac{12}{13}}{\frac{5}{13}} \][/tex]

Simplify the fraction:
[tex]\[ \tan x = \frac{12}{13} \times \frac{13}{5} = \frac{12 \times 13}{13 \times 5} = \frac{12}{5} \][/tex]

Thus:
[tex]\[ \tan x = 2.4 \][/tex]

### Final Answers:
[tex]\[ \sin x = \frac{12}{13} \approx 0.9230769230769231 \][/tex]
[tex]\[ \tan x = 2.4 \][/tex]

In summary:
- [tex]\(\sin x \approx 0.923\)[/tex]
- [tex]\(\tan x = 2.4\)[/tex]