3. For the function $f^{\prime}(x):=$ ig $\left(\sin ^{6} x+\cos ^{6} x+\right.$ $a \sin x \cos x)$ to have the domain $\mathbb{R}$, the real number $a$ must satisfy the condition:
Therefore, it is only necessary to determine $a$ such that when $t \in\left[-\frac{1}{2}, \frac{1}{2}\right]$,
$$
-3 t^{2}+a t+1>0 \text {. }
$$
(i) When $t=0$, (1) always holds;
(ii) When $t>0$, (1) $\Leftrightarrow a>3 t-\frac{1}{t}$. Let $f(t)=3 t$ $-\frac{1}{t}$, at this time $f(t)$ is monotonically increasing on $\mathbf{R}^{+}$,
$$
\therefore a>f(t)_{\max } \xlongequal{t \in\left(0, \frac{1}{2}\right]} f\left(\frac{1}{2}\right)=-\frac{1}{2} ;
$$
For $t<0$, let $g(t)=-3 t-\frac{1}{t}$, at this time $g(t)$ is monotonically increasing on $\mathbf{R}^{-}$, and $g(t)_{\text {min }}=$ $-\frac{3}{2}+2=\frac{1}{2}$, i.e., $a<\frac{1}{2}$.
In summary, we know $-\frac{1}{2}<a<\frac{1}{2}$.