Let's solve the given trigonometric equation step by step.

Given equation:

$2 \sin^2 \theta - 1 = 0$

Step 1: Isolate $\sin^2 \theta$

$2 \sin^2 \theta = 1$ $\sin^2 \theta = \frac{1}{2}$

Step 2: Solve for $\sin \theta$

$\sin \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}$

Step 3: Find the general solutions

The sine function equals $\frac{\sqrt{2}}{2}$ at angles: $\theta = 45^\circ, 135^\circ$

And equals $-\frac{\sqrt{2}}{2}$ at: $\theta = 225^\circ, 315^\circ$

Final Answer:

$\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ$

Would you like further clarification or a graphical representation?

Related Questions:

1. How would you solve this equation in radians instead of degrees?
2. How can this equation be solved graphically?
3. What happens if the domain is extended beyond $0^\circ \leq \theta < 360^\circ$?
4. What other trigonometric functions could be used to verify the solution?
5. Can this equation be solved using a unit circle approach?

Tip: Always check if the given angle domain is in degrees or radians before solving trigonometric equations!