Solve the equation for solutions over the interval [0,360) degress

Question

cot theta + 4 csc theta = 6

what is the solution set { }

Answer 1

cotθ + 4cscθ = 6

cosθ / sinθ + 4 / sinθ = 6

cosθ + 4 = 6sinθ

cosθ - 6sinθ = -4

Multiply and divide the left hand side by √[1² + (-6)²] = √37

√37[(1/√37)cosθ - (6/√37)sinθ] = -4

Plot the point (1,6) and draw the right triangle with vertices (0,0), (1,0), and (1,6). Let β be the acute angle of the triangle with vertex at (0,0).

Then, cosβ = 1/√37 and sinβ = 6/√37

So, we have √37[cosβcosθ - sinβsinθ) = -4

√37cos(θ+β) = -4, where β = Cos^-1(1/√37) = 80.54°

cos(θ + 80.54°) = -0.6576

θ + 80.54° = 180° - Cos^-1(0.6576) or 180° + Cos^-1(0.6576)

θ + 80.54° = 131.12° or 228.88°

θ = 50.58° or 148.34°