If in general , the value of sin A is known , but the value of A is not known , then how many values of tan `((A)/(2))` can be calculated ?

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to determine how many values of \(\tan\left(\frac{A}{2}\right)\) can be calculated when the value of \(\sin A\) is known.

### Step-by-Step Solution:

1. **Understanding the relationship between sine and tangent**:
We know that \(\sin A\) can be expressed in terms of \(\tan\left(\frac{A}{2}\right)\) using the identity:
\[ \sin A = \frac{2 \tan\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)} \]

2. **Setting up the equation**:
Let \(x = \tan\left(\frac{A}{2}\right)\). Then, substituting \(x\) into the sine identity gives:
\[ \sin A = \frac{2x}{1 + x^2} \]
If we know the value of \(\sin A\), we can set this equal to a constant \(k\) (where \(k = \sin A\)):
\[ k = \frac{2x}{1 + x^2} \]

3. **Rearranging the equation**:
Rearranging the equation leads to:
\[ k(1 + x^2) = 2x \]
This simplifies to:
\[ kx^2 - 2x + k = 0 \]

4. **Identifying the quadratic equation**:
The equation \(kx^2 - 2x + k = 0\) is a quadratic equation in \(x\) (where \(x = \tan\left(\frac{A}{2}\right)\)).

5. **Finding the number of solutions**:
A quadratic equation can have 0, 1, or 2 real solutions depending on the discriminant. The discriminant \(\Delta\) of the quadratic equation \(Ax^2 + Bx + C = 0\) is given by:
\[ \Delta = B^2 - 4AC \]
For our equation:
- \(A = k\)
- \(B = -2\)
- \(C = k\)

Thus, the discriminant is:
\[ \Delta = (-2)^2 - 4(k)(k) = 4 - 4k^2 \]

6. **Analyzing the discriminant**:
- If \(\Delta > 0\), there are 2 distinct real solutions for \(x\).
- If \(\Delta = 0\), there is 1 real solution (a repeated root).
- If \(\Delta < 0\), there are no real solutions.

Since \(k = \sin A\) can take values in the range \([-1, 1]\), we can analyze the discriminant:
- For \(\sin A = 1\) (maximum), \(\Delta = 4 - 4(1^2) = 0\) (1 solution).
- For \(\sin A = 0\) (minimum), \(\Delta = 4 - 4(0^2) = 4\) (2 solutions).
- For \(\sin A = -1\), \(\Delta = 4 - 4(-1^2) = 0\) (1 solution).

7. **Conclusion**:
Therefore, depending on the value of \(\sin A\), we can conclude that there can be either 1 or 2 values of \(\tan\left(\frac{A}{2}\right)\) calculated. However, since the question asks how many values can be calculated when \(\sin A\) is known, the answer is:
\[ \text{Two values of } \tan\left(\frac{A}{2}\right) \text{ can be calculated.} \]