If `tan theta + tan (theta + pi/3) + tan ( theta + (2pi)/3) = 3`, then `theta=`

Options

A. `(2n + 1) pi/12`
B. `n pi pm pi/3`
C. `(4n + 1) pi/12`
D. none

Answer

To solve the equation \( \tan \theta + \tan \left( \theta + \frac{\pi}{3} \right) + \tan \left( \theta + \frac{2\pi}{3} \right) = 3 \), we can follow these steps:

### Step 1: Use the tangent addition formula
We know that:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
We will apply this formula to the terms \( \tan \left( \theta + \frac{\pi}{3} \right) \) and \( \tan \left( \theta + \frac{2\pi}{3} \right) \).

### Step 2: Calculate \( \tan \left( \theta + \frac{\pi}{3} \right) \)
Using the formula:
\[ \tan \left( \theta + \frac{\pi}{3} \right) = \frac{\tan \theta + \tan \frac{\pi}{3}}{1 - \tan \theta \tan \frac{\pi}{3}} = \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} \]

### Step 3: Calculate \( \tan \left( \theta + \frac{2\pi}{3} \right) \)
Similarly,
\[ \tan \left( \theta + \frac{2\pi}{3} \right) = \frac{\tan \theta + \tan \frac{2\pi}{3}}{1 - \tan \theta \tan \frac{2\pi}{3}} = \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \sqrt{3}} \]

### Step 4: Substitute back into the original equation
Now, substituting these back into the equation:
\[ \tan \theta + \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} + \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \sqrt{3}} = 3 \]

### Step 5: Combine the terms
Let \( x = \tan \theta \). The equation becomes:
\[ x + \frac{x + \sqrt{3}}{1 - x\sqrt{3}} + \frac{x - \sqrt{3}}{1 + x\sqrt{3}} = 3 \]

### Step 6: Find a common denominator
The common denominator for the fractions is \( (1 - x\sqrt{3})(1 + x\sqrt{3}) \). We can simplify the equation by multiplying through by this common denominator.

### Step 7: Simplify the equation
After simplification, we will have a cubic equation in terms of \( x \):
\[ 3(1 - 3x^2) = 9x - 3x^3 \]
Rearranging gives:
\[ 3x^3 - 9x + 3 = 0 \]
Dividing through by 3:
\[ x^3 - 3x + 1 = 0 \]

### Step 8: Solve the cubic equation
Using the Rational Root Theorem or synthetic division, we can find that \( x = 1 \) is a root. Thus, we can factor the cubic:
\[ (x - 1)(x^2 + x - 1) = 0 \]
The quadratic can be solved using the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{5}}{2} \]

### Step 9: Find the values of \( \theta \)
Since \( x = \tan \theta \), we have:
1. \( \tan \theta = 1 \) gives \( \theta = \frac{\pi}{4} + n\pi \)
2. \( \tan \theta = \frac{-1 + \sqrt{5}}{2} \) and \( \tan \theta = \frac{-1 - \sqrt{5}}{2} \) give additional angles.

### Final Answer
Thus, the general solutions for \( \theta \) are:
\[ \theta = n\pi + \frac{\pi}{4} \quad \text{and other values from the quadratic} \]