The most general value of `theta` which satisfy both the equation `cos theta =-(1)/(sqrt(2)) and tan theta =1 ` , is

Options

A. `2npi +(5pi)/(4), n in 1`
B. `2npi +(pi)/(4), n in 1`
C. `2npi +(3pi)/(4), n in 1`
D. None of these

Answer

To find the most general value of \( \theta \) that satisfies both equations \( \cos \theta = -\frac{1}{\sqrt{2}} \) and \( \tan \theta = 1 \), we can follow these steps:

### Step 1: Analyze the equations
We have two equations:
1. \( \cos \theta = -\frac{1}{\sqrt{2}} \)
2. \( \tan \theta = 1 \)

### Step 2: Determine the angles from the equations
- For \( \tan \theta = 1 \), we know that \( \theta \) can be \( \frac{\pi}{4} + n\pi \) where \( n \) is any integer. This is because tangent is positive in the first and third quadrants.
- For \( \cos \theta = -\frac{1}{\sqrt{2}} \), the angles that satisfy this are \( \theta = \frac{3\pi}{4} + 2n\pi \) and \( \theta = \frac{5\pi}{4} + 2n\pi \).

### Step 3: Identify the relevant quadrants
Since \( \cos \theta \) is negative and \( \tan \theta \) is positive, we are looking for angles in the third quadrant. The third quadrant is where both sine is negative and cosine is negative, making tangent positive.

### Step 4: Find the common angle
From the analysis, we see that the angle \( \frac{5\pi}{4} \) is in the third quadrant and satisfies both equations:
- \( \tan \frac{5\pi}{4} = 1 \)
- \( \cos \frac{5\pi}{4} = -\frac{1}{\sqrt{2}} \)

### Step 5: Write the general solution
The general solution for \( \theta \) can be expressed as:
\[ \theta = 2n\pi + \frac{5\pi}{4} \quad \text{or} \quad \theta = 2n\pi - \frac{5\pi}{4} \quad \text{where } n \in \mathbb{Z} \]

Thus, the most general value of \( \theta \) that satisfies both equations is:
\[ \theta = 2n\pi + \frac{5\pi}{4} \quad \text{or} \quad \theta = 2n\pi - \frac{5\pi}{4} \]