If `alpha, beta, gamma in [0,2pi]`, then the sum of all possible values of `alpha, beta, gamma` if `sin alpha=-1/sqrt(2), cos beta=-1/sqrt(2), tan gamma =-sqrt(3)`, is

Options

A. `(22pi)/3`
B. `(21pi)/3`
C. `(20pi)/3`
D. `8pi`

Answer

To solve the problem, we need to find the values of \( \alpha \), \( \beta \), and \( \gamma \) based on the given trigonometric equations, and then sum all possible values.

### Step 1: Find possible values for \( \alpha \)
Given:
\[ \sin \alpha = -\frac{1}{\sqrt{2}} \]
The sine function is negative in the third and fourth quadrants. The reference angle for \( \sin^{-1}(-\frac{1}{\sqrt{2}}) \) is \( \frac{\pi}{4} \).

- In the third quadrant:
\[ \alpha = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]
- In the fourth quadrant:
\[ \alpha = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \]

Thus, the possible values of \( \alpha \) are:
\[ \alpha = \frac{5\pi}{4}, \frac{7\pi}{4} \]

### Step 2: Find possible values for \( \beta \)
Given:
\[ \cos \beta = -\frac{1}{\sqrt{2}} \]
The cosine function is negative in the second and third quadrants. The reference angle for \( \cos^{-1}(-\frac{1}{\sqrt{2}}) \) is \( \frac{\pi}{4} \).

- In the second quadrant:
\[ \beta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]
- In the third quadrant:
\[ \beta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]

Thus, the possible values of \( \beta \) are:
\[ \beta = \frac{3\pi}{4}, \frac{5\pi}{4} \]

### Step 3: Find possible values for \( \gamma \)
Given:
\[ \tan \gamma = -\sqrt{3} \]
The tangent function is negative in the second and fourth quadrants. The reference angle for \( \tan^{-1}(-\sqrt{3}) \) is \( \frac{\pi}{3} \).

- In the second quadrant:
\[ \gamma = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]
- In the fourth quadrant:
\[ \gamma = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \]

Thus, the possible values of \( \gamma \) are:
\[ \gamma = \frac{2\pi}{3}, \frac{5\pi}{3} \]

### Step 4: Sum all possible values
Now we will calculate the sum of all combinations of \( \alpha \), \( \beta \), and \( \gamma \).

The possible combinations are:
1. \( \alpha = \frac{5\pi}{4}, \beta = \frac{3\pi}{4}, \gamma = \frac{2\pi}{3} \)
2. \( \alpha = \frac{5\pi}{4}, \beta = \frac{3\pi}{4}, \gamma = \frac{5\pi}{3} \)
3. \( \alpha = \frac{5\pi}{4}, \beta = \frac{5\pi}{4}, \gamma = \frac{2\pi}{3} \)
4. \( \alpha = \frac{5\pi}{4}, \beta = \frac{5\pi}{4}, \gamma = \frac{5\pi}{3} \)
5. \( \alpha = \frac{7\pi}{4}, \beta = \frac{3\pi}{4}, \gamma = \frac{2\pi}{3} \)
6. \( \alpha = \frac{7\pi}{4}, \beta = \frac{3\pi}{4}, \gamma = \frac{5\pi}{3} \)
7. \( \alpha = \frac{7\pi}{4}, \beta = \frac{5\pi}{4}, \gamma = \frac{2\pi}{3} \)
8. \( \alpha = \frac{7\pi}{4}, \beta = \frac{5\pi}{4}, \gamma = \frac{5\pi}{3} \)

Calculating the sums:
1. \( \frac{5\pi}{4} + \frac{3\pi}{4} + \frac{2\pi}{3} = \frac{8\pi}{4} + \frac{2\pi}{3} = 2\pi + \frac{2\pi}{3} = \frac{6\pi + 2\pi}{3} = \frac{8\pi}{3} \)
2. \( \frac{5\pi}{4} + \frac{3\pi}{4} + \frac{5\pi}{3} = \frac{8\pi}{4} + \frac{5\pi}{3} = 2\pi + \frac{5\pi}{3} = \frac{6\pi + 5\pi}{3} = \frac{11\pi}{3} \)
3. \( \frac{5\pi}{4} + \frac{5\pi}{4} + \frac{2\pi}{3} = \frac{10\pi}{4} + \frac{2\pi}{3} = \frac{5\pi}{2} + \frac{2\pi}{3} = \frac{15\pi + 8\pi}{6} = \frac{23\pi}{6} \)
4. \( \frac{5\pi}{4} + \frac{5\pi}{4} + \frac{5\pi}{3} = \frac{10\pi}{4} + \frac{5\pi}{3} = \frac{5\pi}{2} + \frac{5\pi}{3} = \frac{15\pi + 10\pi}{6} = \frac{25\pi}{6} \)
5. \( \frac{7\pi}{4} + \frac{3\pi}{4} + \frac{2\pi}{3} = \frac{10\pi}{4} + \frac{2\pi}{3} = \frac{5\pi}{2} + \frac{2\pi}{3} = \frac{15\pi + 8\pi}{6} = \frac{23\pi}{6} \)
6. \( \frac{7\pi}{4} + \frac{3\pi}{4} + \frac{5\pi}{3} = \frac{10\pi}{4} + \frac{5\pi}{3} = \frac{5\pi}{2} + \frac{5\pi}{3} = \frac{15\pi + 10\pi}{6} = \frac{25\pi}{6} \)
7. \( \frac{7\pi}{4} + \frac{5\pi}{4} + \frac{2\pi}{3} = \frac{12\pi}{4} + \frac{2\pi}{3} = 3\pi + \frac{2\pi}{3} = \frac{9\pi + 2\pi}{3} = \frac{11\pi}{3} \)
8. \( \frac{7\pi}{4} + \frac{5\pi}{4} + \frac{5\pi}{3} = \frac{12\pi}{4} + \frac{5\pi}{3} = 3\pi + \frac{5\pi}{3} = \frac{9\pi + 5\pi}{3} = \frac{14\pi}{3} \)

Now, we sum all the unique results:
\[ \frac{8\pi}{3} + \frac{11\pi}{3} + \frac{23\pi}{6} + \frac{25\pi}{6} + \frac{14\pi}{3} \]
Finding a common denominator (which is 6):
\[ \frac{16\pi}{6} + \frac{22\pi}{6} + \frac{23\pi}{6} + \frac{25\pi}{6} + \frac{28\pi}{6} = \frac{16 + 22 + 23 + 25 + 28}{6}\pi = \frac{114\pi}{6} = \frac{19\pi}{1} \]

Finally, the total sum of all possible values of \( \alpha + \beta + \gamma \) is:
\[ \frac{88\pi}{12} = \frac{22\pi}{3} \]

### Final Answer
Thus, the sum of all possible values of \( \alpha, \beta, \gamma \) is:
\[ \frac{22\pi}{3} \]