If `alpha,beta,gamma,delta` are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity `k ,` then the value of `4sin(alpha/2)+3sin(beta/2)+2sin(gamma/2)+sin(delta/2)` is equal to (a)`2sqrt(1-k)` (b) `2sqrt(1+k)` (c)`(sqrt(1-k))/2` (d) none of these

Explanation

To solve the problem, we need to find the value of \(4\sin\left(\frac{\alpha}{2}\right) + 3\sin\left(\frac{\beta}{2}\right) + 2\sin\left(\frac{\gamma}{2}\right) + \sin\left(\frac{\delta}{2}\right)\) given that \(\sin(\alpha) = \sin(\beta) = \sin(\gamma) = \sin(\delta) = k\) and that \(\alpha < \beta < \gamma < \delta\).

### Step-by-Step Solution:

1. **Identify the Angles**:
- Since \(\sin(\alpha) = k\), the smallest positive angle \(\alpha\) is given by \(\alpha = \arcsin(k)\).
- The next angle \(\beta\) can be expressed as \(\beta = \pi - \alpha\) because \(\sin(\beta) = \sin(\alpha)\) and \(\beta\) is in the second quadrant.
- The angle \(\gamma\) can be expressed as \(\gamma = 2\pi + \alpha\) because \(\sin(\gamma) = \sin(\alpha)\) and \(\gamma\) is in the third quadrant.
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