Answer:
θ = 2 π n_1 + π/3 for n_1 element Z
or θ = 2 π n_2 + (5 π)/3 for n_2 element Z
or θ = π n_3 for n_3 element Z
θ = π n_3 for θ element Z and cos(θ)!=0 and n_3 element Z
Explanation:
Solve for θ:
tan(θ) - 2 sin(θ) = 0
Factor sin(θ) from the left hand side:
sin(θ) (sec(θ) - 2) = 0
Split sin(θ) (sec(θ) - 2) into separate parts with additional assumptions.
Assume cos(θ)!=0 from sec(θ):
sec(θ) - 2 = 0 or sin(θ) = 0 for cos(θ)!=0
Add 2 to both sides:
sec(θ) = 2 or sin(θ) = 0 for cos(θ)!=0
Take the reciprocal of both sides:
cos(θ) = 1/2 or sin(θ) = 0 for cos(θ)!=0
Take the inverse cosine of both sides:
θ = 2 π n_1 + π/3 for n_1 element Z or θ = 2 π n_2 + (5 π)/3 for n_2 element Z
or sin(θ) = 0 for cos(θ)!=0
Take the inverse sine of both sides:
θ = 2 π n_1 + π/3 for n_1 element Z
or θ = 2 π n_2 + (5 π)/3 for n_2 element Z
or θ = π n_3 for cos(θ)!=0 and n_3 element Z
The roots θ = π n_3 never violate cos(θ)!=0, which means this assumption can be omitted:
Answer: θ = 2 π n_1 + π/3 for n_1 element Z
or θ = 2 π n_2 + (5 π)/3 for n_2 element Z
or θ = π n_3 for n_3 element Z
Solve for θ over the integers:
tan(θ) - 2 sin(θ) = 0
Factor sin(θ) from the left-hand side:
sin(θ) (sec(θ) - 2) = 0
Split sin(θ) (sec(θ) - 2) into separate parts with additional assumptions.
Assume cos(θ)!=0 from sec(θ):
sec(θ) - 2 = 0 or sin(θ) = 0 for cos(θ)!=0
Add 2 to both sides:
sec(θ) = 2 or sin(θ) = 0 for cos(θ)!=0
Take the reciprocal of both sides:
cos(θ) = 1/2 or sin(θ) = 0 for cos(θ)!=0
Take the inverse cosine of both sides:
θ = 2 π n_1 + π/3 for θ element Z and n_1 element Z or θ = 2 π n_2 + (5 π)/3 for θ element Z and n_2 element Z
or sin(θ) = 0 for cos(θ)!=0
The roots θ = 2 π n_1 + π/3 violate θ element Z for all n_1 element Z:
θ = 2 π n_2 + (5 π)/3 for θ element Z and n_2 element Z
or sin(θ) = 0 for cos(θ)!=0
The roots θ = 2 π n_2 + (5 π)/3 violate θ element Z for all n_2 element Z:
sin(θ) = 0 for cos(θ)!=0
Take the inverse sine of both sides:
Answer: θ = π n_3 for θ element Z and cos(θ)!=0 and n_3 element Z