help with trigonometry plz?

Question

Solve. Give all positive values of the angle between 0º and 360º that will satisfy each. Give any approximate value to the nearest minute only.




3. sin 2 theta = √3/2

7. sin² theta = cos² theta + ½

10. tan (x + 15º) = 3 tan x

Answer 1

3. sin 2θ = √3/2
2θ = 60°,120°,420°,480°
θ = 30°, 60°,210°,240°

7. sin²θ = cos²θ + ½
sin²θ - cos²θ = ½
cos²θ - sin²θ = -½
cos(2θ) = -½
2θ = 120°,240°,480°,600°
θ = 60°,120°,240°,300°

10. tan(x + 15º) = 3 tan x
tan(x + 15º) = (tanx + tan15°)/{1 - (tanx)(tan15°)} = 3tanx
tanx + tan15° = 3tanx - 3(tan²x)(tan15°)
0 = 2tanx - tan15° - 3(tan²x)(tan15°)
3(tan15°)(tan²x) - 2tanx + tan15° = 0
tanx = {2 ± √(4 - 12tan²15°)}/{6(tan15°)}
tanx = {2 ± √(4 - 12(2 - √3)²}/{6(2 - √3)}
tanx = {2 ± √(48√3 - 80}/{6(2 - √3)}
tanx = {2 ± 4√(3√3 - 5}/{6(2 - √3)}
tanx = {1 + 2√(3√3 - 5}/{3(2 - √3)} ≈ 2.3459447
The other solution is rejected.

x = 66.913024°, 246.913024°,