If `Sinx=(4)/(5)`, then `Secx+Tanx=?`

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-by-Step Solution

1. **Understanding the Given Information**:
We know that \( \sin x = \frac{4}{5} \). This means that in a right triangle, the length of the opposite side (perpendicular) to angle \( x \) is 4, and the length of the hypotenuse is 5.

2. **Finding the Length of the Base**:
We can use the Pythagorean theorem to find the length of the adjacent side (base). The Pythagorean theorem states:
\[ h^2 = p^2 + b^2 \]
where \( h \) is the hypotenuse, \( p \) is the opposite side, and \( b \) is the adjacent side. Plugging in the values:
\[ 5^2 = 4^2 + b^2 \]
This simplifies to:
\[ 25 = 16 + b^2 \]
Therefore:
\[ b^2 = 25 - 16 = 9 \]
Taking the square root gives us:
\[ b = 3 \]

3. **Calculating \( \sec x \) and \( \tan x \)**:
- The secant function is defined as:
\[ \sec x = \frac{h}{b} = \frac{5}{3} \]
- The tangent function is defined as:
\[ \tan x = \frac{p}{b} = \frac{4}{3} \]

4. **Finding \( \sec x + \tan x \)**:
Now we can 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, \( \sec x + \tan x = 3 \).