`1/2tan^(- 1)(12/5)` is equal to

Options

A. `tan^(-1)(3/2)`
B. `tan^(-1)(2/3)`
C. `tan^(-1)(3/4)`
D. `tan^(-1)(7/17)`

Answer

To solve the equation \( \frac{1}{2} \tan^{-1} \left( \frac{12}{5} \right) \), we will follow these steps:

### Step 1: Set up the equation
Let:
\[ \theta = \frac{1}{2} \tan^{-1} \left( \frac{12}{5} \right) \]
This implies:
\[ \tan(2\theta) = \frac{12}{5} \]

### Step 2: Use the double angle formula for tangent
The double angle formula for tangent states that:
\[ \tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting this into our equation gives:
\[ \frac{12}{5} = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying yields:
\[ 12(1 - \tan^2(\theta)) = 10 \tan(\theta) \]
Expanding this gives:
\[ 12 - 12 \tan^2(\theta) = 10 \tan(\theta) \]

### Step 4: Rearrange into a quadratic equation
Rearranging the equation results in:
\[ 12 \tan^2(\theta) + 10 \tan(\theta) - 12 = 0 \]

### Step 5: Use the quadratic formula
We can apply the quadratic formula \( \tan(\theta) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 12 \), \( b = 10 \), and \( c = -12 \):
\[ \tan(\theta) = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 12 \cdot (-12)}}{2 \cdot 12} \]
Calculating the discriminant:
\[ = \frac{-10 \pm \sqrt{100 + 576}}{24} \]
\[ = \frac{-10 \pm \sqrt{676}}{24} \]
\[ = \frac{-10 \pm 26}{24} \]

### Step 6: Calculate the two possible values for \( \tan(\theta) \)
Calculating the two possibilities:
1. \( \tan(\theta) = \frac{16}{24} = \frac{2}{3} \)
2. \( \tan(\theta) = \frac{-36}{24} = -\frac{3}{2} \)

### Step 7: Find the angles
Thus, we have:
1. \( \theta = \tan^{-1} \left( \frac{2}{3} \right) \)
2. \( \theta = \tan^{-1} \left( -\frac{3}{2} \right) \)

### Final Result
The two possible values for \( \theta \) are:
\[ \theta = \tan^{-1} \left( \frac{2}{3} \right) \quad \text{or} \quad \theta = \tan^{-1} \left( -\frac{3}{2} \right) \]