If `sin theta=(21)/(29)`, find the value of `sec theta+tan theta`, if `theta` lies between 0 and `pi//2`:

Options

A. 1
B. `1/2`
C. `5/2`
D. none of these

Answer

To solve the problem, we need to find the value of \( \sec \theta + \tan \theta \) given that \( \sin \theta = \frac{21}{29} \) and \( \theta \) lies between \( 0 \) and \( \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Identify the values:**
- We are given \( \sin \theta = \frac{21}{29} \).
- In a right triangle, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \). Here, the opposite side (perpendicular) is 21 and the hypotenuse is 29.

2. **Use the Pythagorean theorem to find the base (adjacent side):**
- According to the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
- Plugging in the values:
\[ 29^2 = 21^2 + \text{adjacent}^2 \]
- Calculate \( 29^2 \) and \( 21^2 \):
\[ 841 = 441 + \text{adjacent}^2 \]
- Rearranging gives:
\[ \text{adjacent}^2 = 841 - 441 = 400 \]
- Taking the square root:
\[ \text{adjacent} = \sqrt{400} = 20 \]

3. **Calculate \( \sec \theta \) and \( \tan \theta \):**
- \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{29}{20} \)
- \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{20} \)

4. **Find \( \sec \theta + \tan \theta \):**
- Now, we can add the two values:
\[ \sec \theta + \tan \theta = \frac{29}{20} + \frac{21}{20} = \frac{29 + 21}{20} = \frac{50}{20} = \frac{5}{2} \]

### Final Answer:
Thus, the value of \( \sec \theta + \tan \theta \) is \( \frac{5}{2} \).

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