If `12 cot^(2) theta-31 cosec theta +32=0`, then the value of `sin theta` is

Options

A. `(3)/(4)" or "1`
B. `(2)/(3)" or "-(2)/(3)`
C. `(4)/(5)" or "(3)/(4)`
D. `pm (1)/(2)`

Answer

To solve the equation \( 12 \cot^2 \theta - 31 \csc \theta + 32 = 0 \) and find the value of \( \sin \theta \), we can follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that:
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]
Substituting these into the equation gives:
\[ 12 \left(\frac{\cos^2 \theta}{\sin^2 \theta}\right) - 31 \left(\frac{1}{\sin \theta}\right) + 32 = 0 \]

### Step 2: Multiply through by \(\sin^2 \theta\) to eliminate the denominators
Multiplying the entire equation by \(\sin^2 \theta\) results in:
\[ 12 \cos^2 \theta - 31 \sin \theta + 32 \sin^2 \theta = 0 \]

### Step 3: Use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\)
Substituting \(\cos^2 \theta\) gives:
\[ 12 (1 - \sin^2 \theta) - 31 \sin \theta + 32 \sin^2 \theta = 0 \]
This simplifies to:
\[ 12 - 12 \sin^2 \theta - 31 \sin \theta + 32 \sin^2 \theta = 0 \]
Combining like terms results in:
\[ (32 - 12) \sin^2 \theta - 31 \sin \theta + 12 = 0 \]
or,
\[ 20 \sin^2 \theta - 31 \sin \theta + 12 = 0 \]

### Step 4: Solve the quadratic equation
Now we have a quadratic equation in terms of \(\sin \theta\):
\[ 20x^2 - 31x + 12 = 0 \]
where \(x = \sin \theta\). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 20\), \(b = -31\), and \(c = 12\). Plugging in these values gives:
\[ x = \frac{31 \pm \sqrt{(-31)^2 - 4 \cdot 20 \cdot 12}}{2 \cdot 20} \]
Calculating the discriminant:
\[ (-31)^2 - 4 \cdot 20 \cdot 12 = 961 - 960 = 1 \]
Thus, we have:
\[ x = \frac{31 \pm 1}{40} \]
Calculating the two possible values:
\[ x_1 = \frac{32}{40} = \frac{4}{5}, \quad x_2 = \frac{30}{40} = \frac{3}{4} \]

### Step 5: Conclusion
The possible values for \(\sin \theta\) are:
\[ \sin \theta = \frac{4}{5} \quad \text{or} \quad \sin \theta = \frac{3}{4} \]