If ` sec theta - tan theta = 1/2,` then `theta` lies in the

Options

A. first quadrant
B. second quadrant
C. third quadrant
D. fourth quadrant

Answer

To solve the problem where \( \sec \theta - \tan \theta = \frac{1}{2} \) and determine which quadrant \( \theta \) lies in, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sec \theta - \tan \theta = \frac{1}{2} \]

### Step 2: Use the identity
Recall the identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 3: Substitute the value
We can substitute \( \sec \theta - \tan \theta \) with \( \frac{1}{2} \):
\[ \left(\frac{1}{2}\right)(\sec \theta + \tan \theta) = 1 \]

### Step 4: Solve for \( \sec \theta + \tan \theta \)
Now, we can solve for \( \sec \theta + \tan \theta \):
\[ \sec \theta + \tan \theta = 2 \]

### Step 5: Set up the system of equations
Now we have two equations:
1. \( \sec \theta - \tan \theta = \frac{1}{2} \) (Equation 1)
2. \( \sec \theta + \tan \theta = 2 \) (Equation 2)

### Step 6: Add the equations
Adding Equation 1 and Equation 2:
\[ (\sec \theta - \tan \theta) + (\sec \theta + \tan \theta) = \frac{1}{2} + 2 \]
This simplifies to:
\[ 2 \sec \theta = \frac{5}{2} \]
Thus,
\[ \sec \theta = \frac{5}{4} \]

### Step 7: Find \( \cos \theta \)
Since \( \sec \theta = \frac{1}{\cos \theta} \):
\[ \cos \theta = \frac{4}{5} \]

### Step 8: Find \( \tan \theta \)
Now, substituting \( \sec \theta \) back into Equation 1 to find \( \tan \theta \):
\[ \frac{5}{4} - \tan \theta = \frac{1}{2} \]
Rearranging gives:
\[ \tan \theta = \frac{5}{4} - \frac{1}{2} = \frac{5}{4} - \frac{2}{4} = \frac{3}{4} \]

### Step 9: Determine the quadrant
Both \( \sec \theta \) and \( \tan \theta \) are positive:
- \( \sec \theta = \frac{5}{4} > 0 \)
- \( \tan \theta = \frac{3}{4} > 0 \)

This indicates that \( \theta \) lies in the **first quadrant**.

### Final Answer
Thus, \( \theta \) lies in the **first quadrant**.

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