If `0^@ lt A lt 90^@` and sin A=`4/5` , find the value of tan A + sec A ?

Answer

To solve the problem, we need to find the value of \( \tan A + \sec A \) given that \( \sin A = \frac{4}{5} \) and \( 0^\circ < A < 90^\circ \).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
Since \( \sin A = \frac{4}{5} \), we can interpret this in terms of a right triangle. Here, the opposite side (perpendicular) to angle \( A \) is 4 units, and the hypotenuse is 5 units.

2. **Finding the Base**:
To find the adjacent side (base), we can use the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{perpendicular}^2 + \text{base}^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + \text{base}^2 \]
This simplifies to:
\[ 25 = 16 + \text{base}^2 \]
Rearranging gives:
\[ \text{base}^2 = 25 - 16 = 9 \]
Taking the square root:
\[ \text{base} = \sqrt{9} = 3 \]

3. **Calculating \( \tan A \)**:
The tangent of angle \( A \) is given by:
\[ \tan A = \frac{\text{perpendicular}}{\text{base}} = \frac{4}{3} \]

4. **Calculating \( \sec A \)**:
The secant of angle \( A \) is the reciprocal of the cosine, which can be calculated as:
\[ \sec A = \frac{\text{hypotenuse}}{\text{base}} = \frac{5}{3} \]

5. **Finding \( \tan A + \sec A \)**:
Now we can add \( \tan A \) and \( \sec A \):
\[ \tan A + \sec A = \frac{4}{3} + \frac{5}{3} = \frac{4 + 5}{3} = \frac{9}{3} = 3 \]

### Final Answer:
\[ \tan A + \sec A = 3 \]