posted by Stacey grahaM on .
Solve the equation in the interval [0•,360•]
Csc theta = 1+cot theta
multipy both sides by sintheta
This is only possible at Theta=0, 90, 360
Reduce to sin and cosines:
Thus, if θ ≠ 0,π 2π ...
Taking advantage of symmetry about π/4, where sinπ/4=cosπ/4, substitute θ=φ-π/4:
Expanding by sum/difference formulae,
sinφcosπ/4-cosφsinπ/4 + cosφcosπ/4+sinφsinπ/4=1
Since sinπ/4=cosπ/4, we cancel terms in cosφ to get
θ=0 or π/2
The first value has been rejected since the beginning, so θ=π/2.
φ=arcsin(sqrt(2)/2)=π/4 or 3π/4 ± 2kπ
θ=φ-π/4=0 or π/2 ±2kπ
Since θ=0 has been rejected since the beginning, we are left with
θ=π/2 (for solution between 0 and 360)