Solve the trigonometric equation: (2theta - 30^) - ((2theta)/(3)) = 0

The given equation is:

$$\sin(2\theta - 30^\circ) - \cos\left(\frac{2\theta}{3}\right) = 0$$

Rearrange the equation:

$$\sin(2\theta - 30^\circ) = \cos\left(\frac{2\theta}{3}\right)$$

Apply this identity to the right side of the equation:

$$\cos\left(\frac{2\theta}{3}\right) = \sin\left(90^\circ - \frac{2\theta}{3}\right)$$

Substitute this back into the equation:

$$\sin(2\theta - 30^\circ) = \sin\left(90^\circ - \frac{2\theta}{3}\right)$$

The general solutions for $\sin A = \sin B$ are $A = B + 360^\circ n$ or $A = (180^\circ - B) + 360^\circ n$, where $n$ is an integer.

Case 1: $2\theta - 30^\circ = 90^\circ - \frac{2\theta}{3} + 360^\circ n$
Combine terms with $\theta$:

$$2\theta + \frac{2\theta}{3} = 90^\circ + 30^\circ + 360^\circ n$$ $$\frac{6\theta + 2\theta}{3} = 120^\circ + 360^\circ n$$ $$\frac{8\theta}{3} = 120^\circ + 360^\circ n$$

Multiply by 3:

$$8\theta = 3(120^\circ + 360^\circ n)$$ $$8\theta = 360^\circ + 1080^\circ n$$

Divide by 8:

$$\theta = \frac{360^\circ}{8} + \frac{1080^\circ}{8} n$$ $$\theta = 45^\circ + 135^\circ n$$

For $n=0$, $\theta = 45^\circ$.
For $n=1$, $\theta = 45^\circ + 135^\circ = 180^\circ$.
For $n=2$, $\theta = 45^\circ + 270^\circ = 315^\circ$.

Case 2: $2\theta - 30^\circ = (180^\circ - (90^\circ - \frac{2\theta}{3})) + 360^\circ n$

$$2\theta - 30^\circ = 180^\circ - 90^\circ + \frac{2\theta}{3} + 360^\circ n$$ $$2\theta - 30^\circ = 90^\circ + \frac{2\theta}{3} + 360^\circ n$$

Combine terms with $\theta$:

$$2\theta - \frac{2\theta}{3} = 90^\circ + 30^\circ + 360^\circ n$$ $$\frac{6\theta - 2\theta}{3} = 120^\circ + 360^\circ n$$ $$\frac{4\theta}{3} = 120^\circ + 360^\circ n$$

Multiply by 3:

$$4\theta = 3(120^\circ + 360^\circ n)$$ $$4\theta = 360^\circ + 1080^\circ n$$

Divide by 4:

$$\theta = \frac{360^\circ}{4} + \frac{1080^\circ}{4} n$$ $$\theta = 90^\circ + 270^\circ n$$

For $n=0$, $\theta = 90^\circ$.

Assuming the solutions are required in the range $0^\circ \le \theta < 360^\circ$, the values for $\theta$ are:
$\boxed{\theta = 45^\circ, 90^\circ, 180^\circ, 315