If `tan A = (3)/(4)`, find the value of `(1)/( sin A) + (1)/( cos A)`

Answer

To solve the problem where \( \tan A = \frac{3}{4} \) and we need to find the value of \( \frac{1}{\sin A} + \frac{1}{\cos A} \), we can follow these steps:

### Step 1: Understand the relationship of tangent
Given \( \tan A = \frac{3}{4} \), we can interpret this in terms of a right triangle. Here, the opposite side (perpendicular) to angle \( A \) is 3 units, and the adjacent side (base) is 4 units.

### Step 2: Draw a right triangle
Construct a right triangle where:
- The opposite side (perpendicular) to angle \( A \) is 3.
- The adjacent side (base) to angle \( A \) is 4.

### Step 3: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
\[ \text{Hypotenuse}^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
\[ \text{Hypotenuse} = \sqrt{25} = 5 \]

### Step 4: Find sine and cosine
Now we can find \( \sin A \) and \( \cos A \):
\[ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5} \]
\[ \cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} \]

### Step 5: Calculate \( \frac{1}{\sin A} + \frac{1}{\cos A} \)
Now we substitute the values of \( \sin A \) and \( \cos A \):
\[ \frac{1}{\sin A} + \frac{1}{\cos A} = \frac{1}{\frac{3}{5}} + \frac{1}{\frac{4}{5}} \]
This simplifies to:
\[ = \frac{5}{3} + \frac{5}{4} \]

### Step 6: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12:
\[ \frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12} \]
\[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \]

### Step 7: Add the fractions
Now we can add the two fractions:
\[ \frac{20}{12} + \frac{15}{12} = \frac{20 + 15}{12} = \frac{35}{12} \]

### Final Answer
Thus, the value of \( \frac{1}{\sin A} + \frac{1}{\cos A} \) is \( \frac{35}{12} \).

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