Question

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(3)
7. 已知為銳角,(1)Tan0= Cose,求Sine (2)
1
1
=
= 3, Sin0+ Cos
Sin Cos
=
2(1 + Tane), Tano ·
Ans : (1)
√5-1
(2)
Cos²0
2
3
√15 (3)1+√2

Answer 1

✨ 最佳解答 ✨

qn

(1) sinθ/cosθ = cosθ
sinθ = cos²θ
sinθ = 1 - sin²θ
sin²θ + sinθ - 1 = 0
sinθ = [-1±√5]/2
但 θ 為銳角 ⇒ sinθ > 0
sinθ = [√5-1]/2

(2) 3sinθcosθ = 1
因為 (sinθ + cosθ)² = sin²θ + 2sinθcosθ + cos²θ
= 1 + 2sinθcosθ
所以 sinθcosθ = ½[(sinθ + cosθ)² - 1]
(3/2)[(sinθ + cosθ)² - 1] = 1
(sinθ + cosθ)² = 5/3
sinθ + cosθ = ±√15/3
但 θ 為銳角 ⇒ sinθ > 0, cosθ > 0
sinθ + cosθ = √15/3

(3) 1/cos²θ = (sin²θ + cos²θ)/cos²θ
= tan²θ + 1
⇒ tan²θ + 1 = 2 + 2tanθ
tan²θ - 2tanθ - 1 = 0
tanθ = [2±√8]/2 = 1±√2
但 θ 為銳角 ⇒ tanθ > 0
tanθ = 1+√2