If cos `theta=12/13` , and `(3pi)/2 le theta lt 2pi` , then tan `theta` = ?

Options

A. `-5/13`
B. `-12/5`
C. `-5/12`
D. `5/12`

Answer

To find the value of \( \tan \theta \) given that \( \cos \theta = \frac{12}{13} \) and \( \frac{3\pi}{2} \leq \theta < 2\pi \), we can follow these steps:

### Step 1: Understand the given information
We know that \( \cos \theta = \frac{12}{13} \). This means that in a right triangle, the adjacent side (base) to angle \( \theta \) is 12 and the hypotenuse is 13.

### Step 2: Use the Pythagorean theorem to find the opposite side
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2 \]
Let \( p \) be the length of the opposite side. Then we have:
\[ 13^2 = 12^2 + p^2 \]
Calculating this gives:
\[ 169 = 144 + p^2 \]
\[ p^2 = 169 - 144 = 25 \]
Taking the square root:
\[ p = \sqrt{25} = 5 \]

### Step 3: Determine the sign of the opposite side
Since \( \theta \) is in the interval \( \frac{3\pi}{2} \leq \theta < 2\pi \) (the fourth quadrant), the sine (and thus the opposite side) is negative. Therefore, the opposite side is:
\[ p = -5 \]

### Step 4: Calculate \( \tan \theta \)
The tangent of an angle is given by the ratio of the opposite side to the adjacent side:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{p}{12} = \frac{-5}{12} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = -\frac{5}{12} \]

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