Solve the following equation for theta (θ) over the given domain:

√3 sin 2θ = cos 2θ

Domain: 0 < θ < 360

tan 2 theta = 1/sqrt 3

2 theta between 0 and 720 degrees
2 theta = 30 degrees is a solution
2 theta = 210 degrees is a solution
2 theta = 30+360 = 390 is a solution
2 theta = 210 + 360 = 570 is a solution
so try
15, 105, 195, 285 degrees for theta

To solve the equation √3 sin 2θ = cos 2θ over the given domain 0 < θ < 360, we can use some trigonometric identities.

1. Start by rewriting the equation using the double angle formulas for sine and cosine:
√3 (2sinθcosθ) = (2cos^2θ - 1)

2. Simplify the equation:
2√3 sinθcosθ = 2cos^2θ - 1

3. Rearrange the terms to one side to obtain a quadratic-like equation:
2cos^2θ - 2√3 sinθcosθ - 1 = 0

4. Make use of the Pythagorean identity sin^2θ + cos^2θ = 1:
2(1 - sin^2θ) - 2√3 sinθcosθ - 1 = 0

5. Replace cosθ with its equivalent expression using sinθ:
2(1 - sin^2θ) - 2√3 sinθ(√(1 - sin^2θ)) - 1 = 0

6. Simplify the equation:
2 - 2sin^2θ - 2√3sin^2θ - 1 = 0

7. Combine like terms:
-4√3sin^2θ - 2sin^2θ + 1 = 0

8. Simplify further:
-2(2√3 + 1)sin^2θ + 1 = 0

9. Solve this quadratic equation for sinθ:
(2√3 + 1)sin^2θ = 1

10. Divide both sides by (2√3 + 1):
sin^2θ = 1 / (2√3 + 1)

11. Take the square root of both sides:
sinθ = ± √(1 / (2√3 + 1))

12. Use a calculator or math software to evaluate the value:
sinθ ≈ ±0.317

13. Since we are looking for solutions over the domain 0 < θ < 360, we need to find the values of θ that correspond to the calculated sinθ values. Using an inverse sine function or a calculator, we can find the angles for which sinθ ≈ ±0.317:
θ ≈ 18.19°, 161.81°, 198.19°, 341.81°

So, the solutions to the equation √3 sin 2θ = cos 2θ in the domain 0 < θ < 360 are approximately θ = 18.19°, 161.81°, 198.19°, and 341.81°.