5. Find the set of values of the expression $\frac{a \cos x + b \sin x + c}{\sqrt{a^{2} + b^{2} + c^{2}}}$, where $x, a, b, c$ are arbitrary numbers such that $a^{2} + b^{2} + c^{2} \neq 0$. (20 points)
Solution: Let $\vec{n}=(a ; b ; c), \vec{m}=(\cos x ; \sin x ; 1)$. Then $|\vec{n}|=\sqrt{a^{2}+b^{2}+c^{2}}, \quad|\vec{m}|=\sqrt{2}$. Consider the function $f(x, a, b, c)=\frac{a \cos x+b \sin x+c}{\sqrt{a^{2}+b^{2}+c^{2}}}=\frac{\vec{n} \cdot \vec{m}}{|\vec{n}|}=|\vec{m}| \cos \alpha=\sqrt{2} \cos \alpha$, where $\alpha-$ is the angle between the vectors $\vec{n}$ and $\vec{m}$. The set of values of the function $f$ is the interval $E=[-\sqrt{2} ; \sqrt{2}]$.

Answer: $E=[-\sqrt{2} ; \sqrt{2}]$.