Question

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SunT
15.若
1+Tane
1-Tane
=3+2V2,求Sine+ Cose。
Ans:
- 108 -
√√3+√6
3

Answer 1

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qn

1 + tanθ = (3 + 2√2)(1 - tanθ)
1 + tanθ = (3 + 2√2) - (3 + 2√2)tanθ
(4 + 2√2)tanθ = 2 + 2√2
tanθ = (2 + 2√2) / (4 + 2√2) = (1 + √2) / (2 + √2)

(1 + √2)² + (2 + √2)² = (3 + 2√2) + (6 + 4√2) = 9 + 6√2
= 9 + 2√18 = (√3 + √6)²
(直角三角形斜邊)

sinθ = (1 + √2) / (√3 + √6)
cosθ = (2 + √2) / (√3 + √6)

sinθ + cosθ = (3 + 2√2) / (√3 + √6)
= (3 + 2√2)(√6 - √3) / (6 - 3)
= (3√6 - 3√3 + 4√3 - 2√6) / 3
= (√3 + √6)/3

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