lf $\alpha,\beta,\gamma,\delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $\mathrm{K}$ then value of $4\displaystyle \sin(\frac{\alpha}{2})+3\sin(\frac{\beta}{2})+2\sin(\frac{\gamma}{2})+\sin(\frac{\delta}{2})$ equals

Given sin(alpha) = sin(beta) = sin(gamma) = sin(delta) = K, and alpha, beta, gamma, delta are in ascending order, we have alpha = arcsin(K), beta = pi - arcsin(K), gamma = 2pi + arcsin(K), delta = 3pi - arcsin(K). Substituting these into the expression and using trigonometric identities simplifies the result to 2*sqrt(1+K).