Let `(sin a) x^(2) + (sin a) x + 1 - cos a = 0`. The set of values of a for which roots of this equation are real and distinct, is

Options

A. `(0, 2 tan^(-1)(1)/(4))`
B. `(o, (2pi)/(3))`
C. `(0, pi)`
D. `(0, 2pi)`

Explanation

To solve the equation \( (\sin a) x^2 + (\sin a) x + 1 - \cos a = 0 \) for the values of \( a \) such that the roots are real and distinct, we need to analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The quadratic equation can be expressed in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = \sin a \)
- \( b = \sin a \)
- \( c = 1 - \cos a \)

2. **Calculate the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (\sin a)^2 - 4(\sin a)(1 - \cos a) \]

3. **Simplify the discriminant**:
Expanding the discriminant:
\[ D = \sin^2 a - 4\sin a + 4\sin a \cos a \]
We can rewrite \( \sin^2 a \) using the Pythagorean identity:
\[ \sin^2 a = 1 - \cos^2 a \]
Thus, we have:
\[ D = (1 - \cos^2 a) - 4\sin a + 4\sin a \cos a \]
Rearranging gives:
\[ D = 1 - \cos^2 a + 4\sin a \cos a - 4\sin a \]

4. **Factor out common terms**:
Notice that we can factor out \( 1 - \cos a \):
\[ D = (1 - \cos a)(1 + \cos a + 4\sin a) \]

5. **Set the discriminant greater than zero**:
For the roots to be real and distinct, we need:
\[ D > 0 \implies (1 - \cos a)(1 + \cos a + 4\sin a) > 0 \]

6. **Analyze the factors**:
- The term \( 1 - \cos a \) is always non-negative since \( \cos a \) ranges from -1 to 1. It is zero when \( a = 2n\pi \) (where \( n \) is an integer).
- The term \( 1 + \cos a + 4\sin a \) must also be positive.

7. **Solve the inequality**:
We need to find when:
\[ 1 + \cos a + 4\sin a > 0 \]
This can be rewritten in terms of \( \tan \frac{a}{2} \):
\[ 1 + 2\cos^2 \frac{a}{2} + 4 \cdot 2\sin \frac{a}{2} \cos \frac{a}{2} > 0 \]
This leads us to:
\[ 1 - 4\tan \frac{a}{2} > 0 \implies \tan \frac{a}{2} < \frac{1}{4} \]

8. **Find the range of \( a \)**:
The solution to \( \tan \frac{a}{2} < \frac{1}{4} \) gives:
\[ -\frac{\pi}{2} < \frac{a}{2} < \tan^{-1}\left(\frac{1}{4}\right) \]
Thus, multiplying through by 2:
\[ -\pi < a < 2\tan^{-1}\left(\frac{1}{4}\right) \]

### Final Answer:
The set of values of \( a \) for which the roots of the equation are real and distinct is:
\[ a \in \left(-\pi, 2\tan^{-1}\left(\frac{1}{4}\right)\right) \]