If `(log)_(10)sinx+(log)_(10)cosx=-1a n d(log)_(10)(sinx+cosx)=(((log)_(10)n)-1)/2,` then the value of `' n//3'` is_______

Answer

To solve the problem, we start with the two given equations:

1. \( \log_{10}(\sin x) + \log_{10}(\cos x) = -1 \)
2. \( \log_{10}(\sin x + \cos x) = \frac{\log_{10}(n) - 1}{2} \)

### Step 1: Simplify the first equation
Using the property of logarithms that states \( \log_a(m) + \log_a(n) = \log_a(m \cdot n) \), we can combine the logs in the first equation:

\[ \log_{10}(\sin x \cdot \cos x) = -1 \]

### Step 2: Exponentiate to remove the logarithm
To eliminate the logarithm, we exponentiate both sides:

\[ \sin x \cdot \cos x = 10^{-1} = \frac{1}{10} \]

### Step 3: Use the identity for sine and cosine
We know that \( \sin x \cdot \cos x = \frac{1}{2} \sin(2x) \). Therefore, we can write:

\[ \frac{1}{2} \sin(2x) = \frac{1}{10} \]

### Step 4: Solve for \( \sin(2x) \)
Multiplying both sides by 2 gives:

\[ \sin(2x) = \frac{2}{10} = \frac{1}{5} \]

### Step 5: Substitute into the second equation
Now we will use the second equation. We need to express \( \sin x + \cos x \) in terms of \( \sin(2x) \). We know that:

\[ \sin^2 x + \cos^2 x = 1 \]

Let \( t = \sin x + \cos x \). Then:

\[ t^2 = \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + 2\left(\frac{1}{10}\right) = 1 + \frac{1}{5} = \frac{6}{5} \]

Thus,

\[ t = \sqrt{\frac{6}{5}} \]

### Step 6: Substitute \( t \) into the second equation
Now we substitute \( t \) into the second equation:

\[ \log_{10}\left(\sqrt{\frac{6}{5}}\right) = \frac{\log_{10}(n) - 1}{2} \]

### Step 7: Simplify the left side
Using the property of logarithms \( \log_a(m^n) = n \log_a(m) \):

\[ \frac{1}{2} \log_{10}\left(\frac{6}{5}\right) = \frac{\log_{10}(n) - 1}{2} \]

### Step 8: Multiply through by 2
Multiplying through by 2 gives:

\[ \log_{10}\left(\frac{6}{5}\right) = \log_{10}(n) - 1 \]

### Step 9: Rearranging to find \( n \)
Adding 1 to both sides:

\[ \log_{10}(n) = \log_{10}\left(\frac{6}{5}\right) + 1 \]

This can be rewritten as:

\[ \log_{10}(n) = \log_{10}\left(\frac{6}{5} \cdot 10\right) = \log_{10}(12) \]

### Step 10: Solve for \( n \)
Thus, we find:

\[ n = 12 \]

### Step 11: Find \( \frac{n}{3} \)
Finally, we calculate:

\[ \frac{n}{3} = \frac{12}{3} = 4 \]

### Final Answer
The value of \( \frac{n}{3} \) is \( \boxed{4} \).