If `sin theta =(21)/(29) and theta` lies in the second quadrant, find the value of `sec theta + tan theta` is

Options

A. `2/5`
B. `5/2`
C. `-2/5`
D. `-5/2`

Answer

To find the value of \( \sec \theta + \tan \theta \) given that \( \sin \theta = \frac{21}{29} \) and \( \theta \) lies in the second quadrant, we can follow these steps:

### Step 1: Identify the values of sine, cosine, and tangent
Since \( \sin \theta = \frac{21}{29} \), we know that:
- Opposite side (perpendicular) = 21
- Hypotenuse = 29

### Step 2: Use the Pythagorean theorem to find the adjacent side (base)
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the known values:
\[ 29^2 = 21^2 + \text{Adjacent}^2 \]
Calculating:
\[ 841 = 441 + \text{Adjacent}^2 \]
\[ \text{Adjacent}^2 = 841 - 441 = 400 \]
\[ \text{Adjacent} = \sqrt{400} = 20 \]

### Step 3: Determine the values of \( \sec \theta \) and \( \tan \theta \)
In the second quadrant:
- \( \cos \theta \) is negative and \( \tan \theta \) is negative.

Calculating \( \sec \theta \) and \( \tan \theta \):
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{29}{20} \]
Since \( \sec \theta \) is negative in the second quadrant:
\[ \sec \theta = -\frac{29}{20} \]

Calculating \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{21}{20} \]
Since \( \tan \theta \) is also negative in the second quadrant:
\[ \tan \theta = -\frac{21}{20} \]

### Step 4: Calculate \( \sec \theta + \tan \theta \)
Now we can find \( \sec \theta + \tan \theta \):
\[ \sec \theta + \tan \theta = -\frac{29}{20} - \frac{21}{20} \]
Combining the fractions:
\[ \sec \theta + \tan \theta = -\frac{29 + 21}{20} = -\frac{50}{20} = -\frac{5}{2} \]

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