? 發問時間: 科學及數學天文學及太空 · 2 個月前

Please answer the following question?

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  • 2 個月前
    最佳解答

    40.

    (a) 2 cot(φ+35°) = 5

    cot(φ+35°) = 5/2

    tan(φ+35°) = 2/5

    φ+35° = atan(2/5) ± k.180°

           = 21.8° ± k.180°

    φ = -13.2° ± k.180°

    ∴ φ = 166.8° or 346.8°

    (b)(i)

    secθ/(cotθ+tanθ)

       = (1/cosθ)/(cosθ/sinθ + sinθ/cosθ)

       = 1/(cos^2(θ)/sinθ + sinθ)

       = 1/(cos^2(θ)+sin^2(θ))/sinθ)

       = 1/(1/sinθ)

       = sinθ

    (ii) sec(3θ)/(cot(3θ)+tan(3θ)) = -√3/2

    ∴ sin(3θ) = -√3/2

    -3π/2 ≦ 3θ ≦ 3π/2

    ∴ 3θ = -π/3 or 4π/3

    ∴ θ = -π/9 or 4π/9

    41. cosecθ = 3sinθ+cotθ

    ∴ 1/sinθ = 3sinθ+cosθ/sinθ

    0° < θ < 180°,

    ∴ sinθ ≠ 0,

    方程式兩邊同乘以 sinθ, 得

       1 = 3sin^2(θ) + cosθ

         = 3 - 3cos^2(θ) + cosθ

    ∴ 3cos^2(θ) - cos(θ) - 2 = 0

    ∴ cosθ = [1 ± √(1+24)]/6 = 1 or -2/3

    但 0° < θ < 180°, cosθ ≠ 1,

    ∴ cosθ = -2/3

       θ = acos(-2/3) = 131.81°

    42.(i)

    2cos(x)cot(x) + 1 = cot(x) + 2cos(x)

    ∴ 2cos(x)cot(x) - cot(x) - 2cos(x) + 1 = 0

    ∴ (2cos(x)-1)(cot(x)-1) = 0

    cot(x) 有定義, 則 sin(x)≠0.

    方程式兩邊乘以 sin(x), 則得等價方程式:

        (2cos(x)-1)(cos(x)-sin(x)) = 0

    ∴ a = 2, b = 1.

    (ii)

    ∴ cos(x) = 1/2 or cos(x) = sin(x)

    0 < x < π

        cos(x) = 1/2 if and only if x = π/3

        cos(x) = sin(x) if and only if x = π/4

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