F. 4 M2 Maths

Ching

2015-01-26 3:10 am

1.
(a) Prove that tanA=(1+sin2A-cos2A)/(1+sin2A+cos2A)
(b) Hence, find the value of tan(π/8)

2.
(a) Let x = tanθ. Show that (1-x²)/(1+x²)=cos2θ
(b) Using (a), find maximum value of (3+x²)/(1+x²), where x is real.

回答 (2)

用戶2053

2015-01-26 5:23 am

✔ 最佳答案

1a)(1 + sin2A - cos2A) / (1 + sin2A + cos2A)　1 + 2sinAcosA - (1 - 2sin²A)
= ──────────────────
　1 + 2sinAcosA + (2cos²A -1)　2sinA (cosA + sinA)
= ─────────────
　2cosA (sinA + cosA)= tanAb)Putting A = π/8 ,
tan(π/8)
= (1 + sin(π/4) - cos(π/4)) / (1 + sin(π/4) + cos(π/4)) ...... By the result of (a)
= (1 + √2/2 - √2/2) / (1 + √2/2 + √2/2)
= 1 / (1 + √2)
= √2 - 1
2a)(1 - x²) / (1 + x²)
= (1 - tan²θ) / (1 + tan²θ)
= (1 - sin²θ/cos²θ) / (1 + sin²θ/cos²θ)
= (cos²θ - sin²θ) / (cos²θ + sin²θ)
= cos²θ - sin²θ
= cos2θb)(3 + x²) / (1 + x²)
= 2 + (1 - x²) / (1 + x²)
= 2 + cos2θ ...... By (a)
≤ 2 + 1 ...... ∵ - 1 ≤ cos2θ ≤ 1
= 3
∴ The maximum value of (3 + x²) / (1 + x²) = 3.

2015-01-25 21:38:59 補充：
知足常樂真的很有風度!

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知足常樂

2015-01-26 5:34 am

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2015-01-25 21:42:34 補充：
Cheers!

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