Form 4 M2 - Ans.fyi 參考答案

Form 4 M2

2015-01-26 3:03 am

1.
(a) Prove that (1-cos2θ) / (1+cos2θ) = tan²θ
(b) Hence, find the value of tan²15°

2.
(a) Prove that cos4θ=8(cos^4)θ-8cos²θ+1.
(b) Hence, find the value of (cos^4)(π/12)- cos²(π/12)

回答 (1)

少年時

2015-01-26 4:57 am

✔ 最佳答案

1－cos 2θ＝2 sin²θ, 1＋cos 2θ＝2 cos²θ
(1a) LHS＝(1－cos2θ) / (1＋cos2θ)＝(2 sin²θ) / (2 cos²θ)＝tan²θ＝RHS
(1b) tan²15°＝(1－cos 30°) / (1＋cos 30°) ⋯⋯⋯⋯ (from the result in part 1a)＝(1－√3 / 2) / (1＋√3 / 2)＝(2－√3) / (2＋√3)
(2a) LHS＝cos 4θ＝2 cos²2θ－1＝2(2 cos²θ－1)²－1＝8 cos⁴θ－8 cos²θ＋1＝RHS
(2b) cos⁴(π/12)－cos²(π/12)＝(1/8)(8 cos⁴(π/12)－8 cos²(π/12)＋1)－1/8＝(1/8) cos (π/3)－1/8＝(1/8)(1/2)－1/8＝-1/16

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