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, we need to find the value of \( \sin 2\alpha \) given that \( \alpha \) is a root of the equation \( 25 \cos^2 \theta + 5 \cos \theta - 12 = 0 \) and \( \frac{\pi}{2} < \alpha < \pi \).

### Step 1: Solve the quadratic equation for \( \cos \theta \)

The given equation is:
\[ 25 \cos^2 \theta + 5 \cos \theta - 12 = 0 \]
This is a quadratic equation in terms of \( \cos \theta \). We can use the quadratic formula:
\[ \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 25 \), \( b = 5 \), and \( c = -12 \).

### Step 2: Substitute the values into the quadratic formula

Substituting the values:
\[ \cos \theta = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 25 \cdot (-12)}}{2 \cdot 25} \]
Calculating the discriminant:
\[ 5^2 - 4 \cdot 25 \cdot (-12) = 25 + 1200 = 1225 \]
Now substituting back:
\[ \cos \theta = \frac{-5 \pm \sqrt{1225}}{50} \]
Since \( \sqrt{1225} = 35 \), we have:
\[ \cos \theta = \frac{-5 \pm 35}{50} \]

### Step 3: Calculate the two possible values for \( \cos \theta \)

Calculating the two cases:
1. \( \cos \theta = \frac{-5 + 35}{50} = \frac{30}{50} = \frac{3}{5} \)
2. \( \cos \theta = \frac{-5 - 35}{50} = \frac{-40}{50} = -\frac{4}{5} \)

### Step 4: Determine which value is valid for \( \alpha \)

Given that \( \frac{\pi}{2} < \alpha < \pi \), \( \alpha \) is in the second quadrant where \( \cos \theta \) is negative. Therefore, we take:
\[ \cos \alpha = -\frac{4}{5} \]

### Step 5: Find \( \sin \alpha \)

Using the Pythagorean identity:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
Substituting \( \cos \alpha = -\frac{4}{5} \):
\[ \sin^2 \alpha + \left(-\frac{4}{5}\right)^2 = 1 \]
\[ \sin^2 \alpha + \frac{16}{25} = 1 \]
\[ \sin^2 \alpha = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the positive root (since sine is positive in the second quadrant):
\[ \sin \alpha = \frac{3}{5} \]

### Step 6: Calculate \( \sin 2\alpha \)

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

### Final Answer

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

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