a) $\sin x = 3m - 1$

Ta có: $-1 \leq \sin x \leq 1$

$\to -1 \leq 3m - 1 \leq 1$

$\to 0 \leq 3m \leq 2$

$\to 0 \leq m \leq \dfrac{2}{3}$

b) $\sin^4x + \cos^4x = m$

$\to (\sin^2x + \cos^2x)^2 - 2sin^2x\cos^2x = m$

$\to 1 - \dfrac{1}{2}\sin^22x = m$

$\to 1 - \dfrac{1}{4}(1 - \cos4x) = m$

$\to \cos4x = 4m - 3$

Ta có: $-1 \leq \cos4x \leq 1$

$\to -1 \leq 4m - 3\leq 1$

$\to 2 \leq 4m \leq 4$

$\to \dfrac{1}{2} \leq m \leq 2$

c) $\sin2x + (m+1)\cos2x + 3= 0$

Phương trình có nghiệm $\Leftrightarrow 1^2 + (m+1)^2 \geq 3^2$

$\Leftrightarrow (m + 1)^2 \geq 8$

$\Leftrightarrow \left[\begin{array}{l}m + 1 \geq 2\sqrt2\\m + 1 \leq -2\sqrt2\end{array}\right.$

$\Leftrightarrow \left[\begin{array}{l}m \geq 2\sqrt2-1\\m \leq -2\sqrt-1\end{array}\right.$

d) $4(\sin^6x + \cos^6x) + 2m\sin2x\cos2x + 1 = 0$

$\to 4[(\sin^2x + \cos^2x)^2 - 3\sin^2x\cos^2x(\sin^2x+\cos^2x)] + m\sin4x + 1 = 0$

$\to 4 - 3\sin^22x + m\sin4x + 1 =0$

$\to 4 - \dfrac{3}{2}(1 - \cos4x) + m\sin4x + 1 = 0$

$\to 3\cos4x + 2m\sin4x + 7 = 0$

Phương trình có nghiệm $\Leftrightarrow 3^2 + (2m)^2 \geq 7^2$

$\Leftrightarrow 4m^2 \geq 40$

$\Leftrightarrow m^2 \geq 10$

$\Leftrightarrow \left[\begin{array}{l}m \geq \sqrt{10}\\m \leq - \sqrt{10}\end{array}\right.$