Z=-3*cos+2i*sin 1.當角度=4兀/3時 求Z的絕對值 2.若Z所對贈的點在直線X+3Y=0 求(2cos平方角度/2)/根號2*sin(角度+兀/4)的直?

2016-02-14 12:41 pm
更新1:

Z=-3*cos+2i*sin 1.當角度=4兀/3時 求Z的絕對值 2.若Z所對應的點在直線X+3Y=0 求(2cos平方角度/2)/根號2*sin(角度+兀/4)的直?

回答 (1)

2016-02-14 3:01 pm
✔ 最佳答案
1)
Z = -3cos(4π/3) + 2isin(4π/3)
Z = 3cos(π/3) - 2isin(π/3)
Z = 3/2 - i√3
|Z| = √((3/2)² + (-√3)²) = √21/2

2)
Z = -3cosθ + 2isinθ
Z所對應的點是(-3cosθ , 2sinθ) , 故
-3cosθ + 3(2sinθ) = 0
2sinθ = cosθ
4sin²θ = cos²θ
4sin²θ = 1 - sin²θ
sin²θ = 1/5
sinθ = √5/5 或 sinθ = -√5/5

2cos²(θ/2) / (√2 sin(θ+π/4))
= (1+cosθ) / (√2 (sinθ cos(π/4) + sin(π/4) cosθ) )
= (1+cosθ) / (√2 (sinθ √2/2 + √2/2 cosθ) )
= (1+cosθ) / (sinθ + cosθ)
= (1+2sinθ) / (3sinθ)
= 1/(3sinθ) + 2/3
= 1/(3√5/5) + 2/3 或 -1/(3√5/5) + 2/3
= (2+√5)/3 或 (2 -√5)/3


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