If `theta` lies in the second quadrant and `tan theta=(-5)/(12)`, find the value of `(2cos theta)/(1-sin theta)`.

Answer

To solve the problem step by step, we need to find the value of \(\frac{2 \cos \theta}{1 - \sin \theta}\) given that \(\tan \theta = -\frac{5}{12}\) and \(\theta\) lies in the second quadrant.

### Step 1: Understand the given information
We know that:
- \(\tan \theta = \frac{\text{perpendicular}}{\text{base}} = -\frac{5}{12}\)
- Since \(\theta\) is in the second quadrant, \(\sin \theta\) is positive and \(\cos \theta\) is negative.

### Step 2: Set up the triangle
From \(\tan \theta = -\frac{5}{12}\), we can interpret this as:
- Perpendicular (opposite side) = 5
- Base (adjacent side) = 12

However, since we are in the second quadrant, the base will be negative:
- Perpendicular = 5
- Base = -12

### Step 3: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
\[ \text{Hypotenuse}^2 = 5^2 + (-12)^2 = 25 + 144 = 169 \]
\[ \text{Hypotenuse} = \sqrt{169} = 13 \]

### Step 4: Find \(\sin \theta\) and \(\cos \theta\)
Now we can find \(\sin \theta\) and \(\cos \theta\):
\[ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{5}{13} \]
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{-12}{13} \]

### Step 5: Substitute values into the expression
Now we substitute \(\cos \theta\) and \(\sin \theta\) into the expression \(\frac{2 \cos \theta}{1 - \sin \theta}\):
\[ \frac{2 \cos \theta}{1 - \sin \theta} = \frac{2 \left(-\frac{12}{13}\right)}{1 - \frac{5}{13}} \]

### Step 6: Simplify the expression
First, simplify the denominator:
\[ 1 - \frac{5}{13} = \frac{13}{13} - \frac{5}{13} = \frac{8}{13} \]
Now substitute this back into the expression:
\[ \frac{2 \left(-\frac{12}{13}\right)}{\frac{8}{13}} = \frac{-24/13}{8/13} \]
This simplifies to:
\[ \frac{-24}{8} = -3 \]

### Final Answer
Thus, the value of \(\frac{2 \cos \theta}{1 - \sin \theta}\) is \(-3\).

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