A Sine Expression

Using the Euler's identity:

$\begin{aligned} e^{i5\theta} & = \left(e^{i\theta} \right)^5 \\ \cos{5\theta} + i \sin{5\theta} & = (\cos{\theta} + i \sin{\theta})^5 \\ \Rightarrow \sin{5\theta} & = \Im (\color{#3D99F6}{\cos{\theta}} + i \color{#D61F06}{\sin{\theta}})^5 \quad \quad [\text{Let } \color{#3D99F6} {a=\cos{\theta}} \text{ and } \color{#D61F06} {b=\sin{\theta}} ] \\ & = \Im (\color{#3D99F6}{a} + i \color{#D61F06}{b})^5 \\ & = \Im (a^5 + i5a^4b - 10a^3b^2 - i10a^2b^3 + 5ab^4 + ib^5) \\ & = 5a^4b - 10a^2b^3 + b^5 \\ & = 5(1-b^2)^2b - 10(1-b^2)b^3 + b^5 \\ & = 5(1-2b^2+b^4)b - 10b^3 + 10 b^5 + b^5 \\ & = 5b -10b^3+5b^5 - 10b^3 + 10 b^5 + b^5 \\ & = 16b^5 - 20b^3 + 5b \\ & = 16\sin^5{\theta} - 20\sin^3{\theta} + 5\sin{\theta} \end{aligned}$

$\Rightarrow a + b + c -14 = 16+20+5-14 = \boxed{27}$