If sin x =`(4)/(5)` then sec x + Tan x = ?

Options

A. 37/20
B. 31/12
C. 3
D. `1/3`

Answer

To solve the problem where \( \sin x = \frac{4}{5} \) and we need to find \( \sec x + \tan x \), we can follow these steps:

### Step 1: Understand the relationship of sine, cosine, and tangent
We know that:
- \( \sin x = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
- \( \cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
- \( \tan x = \frac{\text{Opposite}}{\text{Adjacent}} \)
- \( \sec x = \frac{1}{\cos x} \)

### Step 2: Identify the sides of the triangle
Given \( \sin x = \frac{4}{5} \):
- Opposite side = 4
- Hypotenuse = 5

### Step 3: Calculate the adjacent side using the Pythagorean theorem
Using the Pythagorean theorem:
\[ \text{Adjacent}^2 + \text{Opposite}^2 = \text{Hypotenuse}^2 \]
Let the adjacent side be \( a \):
\[ a^2 + 4^2 = 5^2 \]
\[ a^2 + 16 = 25 \]
\[ a^2 = 25 - 16 = 9 \]
\[ a = \sqrt{9} = 3 \]

### Step 4: Calculate \( \sec x \) and \( \tan x \)
Now we can find \( \sec x \) and \( \tan x \):
- \( \sec x = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{5}{3} \)
- \( \tan x = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{4}{3} \)

### Step 5: Add \( \sec x \) and \( \tan x \)
Now we add \( \sec x \) and \( \tan x \):
\[ \sec x + \tan x = \frac{5}{3} + \frac{4}{3} = \frac{5 + 4}{3} = \frac{9}{3} = 3 \]

### Final Answer
Thus, \( \sec x + \tan x = 3 \).

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