Trig

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Solve the equation in the interval [0•,360•]
Csc theta = 1+cot theta

  • Trig -

    multipy both sides by sintheta
    1=sinTheta+cosTheta
    This is only possible at Theta=0, 90, 360

  • Trig -

    Reduce to sin and cosines:
    1/sinθ=(sinθ+cosθ)/sinθ

    Thus, if θ ≠ 0,π 2π ...
    we get
    sinθ+cosθ=1

    Taking advantage of symmetry about π/4, where sinπ/4=cosπ/4, substitute θ=φ-π/4:
    sin(φ-π/4)+cos(φ-π/4)=1
    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
    2sinπ/4 sinφ=1
    φ=arcsin(sqrt(2)/2)=±π/4
    θ=0 or π/2
    The first value has been rejected since the beginning, so θ=π/2.

  • Trig - correction -

    from
    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)

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