If f(x) = left| begin{array}{ccc} sin(x + alp | vedclass

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(C) Let $f(x) = \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamm

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$10$

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(C) Let $f(x) = \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \ \sin(\alpha + \beta) & \sin(\beta + \gamma) & \sin(\gamma + \alpha) \end{array} ight|$.To find $f'(x)$,we differentiate the determinant row by row.$f'(x) = \left| \begin{array}{ccc} \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \ \sin(\alpha + \beta) & \sin(\beta + \gamma) & \sin(\gamma + \alpha) \end{array} ight| + \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \ -\sin(x + \alpha) & -\sin(x + \beta) & -\sin(x + \gamma) \ \sin(\alpha + \beta) & \sin(\beta + \gamma) & \sin(\gamma + \alpha) \end{array} ight| + \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \ 0 & 0 & 0 \end{array} ight|$.In the first determinant,row $1$ and row $2$ are identical,so its value is $0$.In the second determinant,row $2$ is $-1$ times row $1$,so its value is $0$.In the third determinant,row $3$ consists of all zeros,so its value is $0$.Thus,$f'(x) = 0 + 0 + 0 = 0$.Since the derivative is zero,$f(x)$ is a constant function.Given $f(10) = 10$,it follows that $f(x) = 10$ for all $x$.Therefore,$f(\pi) = 10$.

If $f(x) = \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \\ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \\ \sin(\alpha + \beta) & \sin(\beta + \gamma) & \sin(\gamma + \alpha) \end{array} \right|$ and $f(10) = 10$,then $f(\pi)$ is equal to

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