sin(θ)/cos(θ) + cos(θ)/sin(θ) = 3
Combine the fractions on the left side by rewriting them with a common denominator.
sin²(θ) / (cos(θ) sin(θ)) + cos²(θ) / (sin(θ) cos(θ)) = 3
(sin²(θ) + cos²(θ)) / (cos(θ) sin(θ)) = 3
Recall the Pythagorean identity, cos²(θ) + sin²(θ) = 1, so that
1 / (cos(θ) sin(θ)) = 3
Recall the double angle identity for sine, sin(2θ) = 2 sin(θ) cos(θ). Then
2 / sin(2θ) = 3
sin(2θ) = 2/3
Take the inverse sine of both sides and solve for θ :
2θ = arcsin(2/3) + 360° n or 2θ = 180° - arcsin(2/3) + 360° n
(where n is any integer)
θ = 1/2 arcsin(2/3) + 180° n or θ = 90° - 1/2 arcsin(2/3) + 180° n
We get a total of 4 solutions between 0° and 360° from both solution sets when n = 0 and n = 1 :
θ = 1/2 arcsin(2/3) ≈ 20.905°
θ = 1/2 arcsin(2/3) + 180° ≈ 200.905°
θ = 90° - 1/2 arcsin(2/3) ≈ 69.095°
θ = 270° - 1/2 arcsin(2/3) ≈ 249.095°