Solve the equation 2\[0.5ex] theta - 3 theta − 2 = 0 for 0^\[0.5ex] theta 360^\[0.5ex] using the t-formula.

Let $t = \tan\left(\frac{\theta}{2}\right)$. The t-formula states that $\sin \theta = \frac{2t}{1+t^2}$ and $\cos \theta = \frac{1-t^2}{1+t^2}$.
Substitute these into the given equation:

$$2\left(\frac{1-t^2}{1+t^2}\right) - 3\left(\frac{2t}{1+t^2}\right) - 2 = 0$$

Multiply the entire equation by $(1+t^2)$ to eliminate the denominators:

$$2(1-t^2) - 3(2t) - 2(1+t^2) = 0$$

Expand the terms:

$$2 - 2t^2 - 6t - 2 - 2t^2 = 0$$

Combine like terms:

$$-4t^2 - 6t = 0$$

Divide by $-2$ to simplify:

$$2t^2 + 3t = 0$$

Factor out $t$:

$$t(2t + 3) = 0$$

This gives two possible values for $t$:

$$t = 0 \quad \text{or} \quad 2t + 3 = 0 \implies t = -\frac{3}{2}$$

Case 1: $t = 0$

$$\tan\left(\frac{\theta}{2}\right) = 0$$

The general solution for $\tan x = 0$ is $x = n \cdot 180