If `alpha` is a root of `25"cos"^(2) theta+ 5"cos" theta-12 = 0, (pi)/(2) lt alpha lt pi, " then sin"2 alpha` is equal to

Options

A. `(24)/(25)`
B. `-(24)/(25)`
C. `(13)/(18)`
D. `-(13)/(18)`

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript, ensuring clarity at each stage.

### Step 1: Solve the Trigonometric Equation
We start with the equation:
\[ 25 \cos^2 \theta + 5 \cos \theta - 12 = 0 \]

This is a quadratic equation in terms of \( \cos \theta \).

### Step 2: Factor the Quadratic Equation
We can rewrite the equation as:
\[ 25 \cos^2 \theta + 20 \cos \theta - 15 \cos \theta - 12 = 0 \]
Grouping the terms gives us:
\[ (25 \cos^2 \theta + 20 \cos \theta) + (-15 \cos \theta - 12) = 0 \]
Factoring out common terms:
\[ 5 \cos \theta (5 \cos \theta + 4) - 3(5 \cos \theta + 4) = 0 \]
This leads to:
\[ (5 \cos \theta - 3)(5 \cos \theta + 4) = 0 \]

### Step 3: Find the Roots
Setting each factor to zero gives us:
1. \( 5 \cos \theta - 3 = 0 \) → \( \cos \theta = \frac{3}{5} \)
2. \( 5 \cos \theta + 4 = 0 \) → \( \cos \theta = -\frac{4}{5} \)

### Step 4: Determine the Acceptable Root
Since we are given that \( \frac{\pi}{2} < \alpha < \pi \), \( \alpha \) is in the second quadrant where cosine is negative. Therefore, we take:
\[ \cos \alpha = -\frac{4}{5} \]

### Step 5: Calculate \( \sin \alpha \)
Using the Pythagorean identity:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
We find \( \sin \alpha \):
\[ \sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left(-\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
Thus,
\[ \sin \alpha = \sqrt{\frac{9}{25}} = \frac{3}{5} \]
Since \( \alpha \) is in the second quadrant, \( \sin \alpha \) is positive.

### Step 6: Calculate \( \sin 2\alpha \)
Using the double angle formula:
\[ \sin 2\alpha = 2 \sin \alpha \cos \alpha \]
Substituting the values we found:
\[ \sin 2\alpha = 2 \left(\frac{3}{5}\right) \left(-\frac{4}{5}\right) = 2 \cdot \frac{3 \cdot -4}{25} = -\frac{24}{25} \]

### Final Answer
Thus, the value of \( \sin 2\alpha \) is:
\[ \sin 2\alpha = -\frac{24}{25} \]

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