中学数学漫谈---扇形之美

2çbçØ}øøm$51 £üà ]õ¬“UG/—ý? —2íã3 iUîÖDÒ�À, GMîmøz, ÔGI— Ì.îƒTA, çF¬Gzmäø¥, ¥|ø _m$v, µ�Öœ1ƒ|òõ, .øIJÖ ýÞ4hV, IA˝˝Ø� −− ¥u�—í m$51� Ê�S,ím$ª¥_y1, .c1Ê Õ$, 61Êq°, 5ó_1¶/=B−V: lzpÊbç,m$íì2: Êø_Æ ,i¦ø¨ÆC, Í(¬¤Cís«õ}� DÆ-©(Ts‘š�, †¤ùš�DÆC Fˇí–�ÿ˚Ñø_m$� Y¤ì2]ª z Þ,픇A/P0Çá��, ��í i�.�¬360�,†��F£¬íø_–� ÿuø_m$; çÍøzm, øw"Ç6uø _m$, N7Ò,J…<Ñ2-í�ɶ} 6uø_m$� m$Ê�S,5ó_1¶? ]�)òÆ VíVÞ�5ó°)íý? BbªJ‚àm $íÞ�V°� lTø--�t�: qm$í š�Ñ r, CÅÑ S, 2-iÑ θ (C�), Þ �Ñ A, † (1) S = rθ (2) A = 1 2 r2θ = 1 2 rS �),Þt�(, BbÿªJà…V°òÆ VíVÞ�: qòÆVí�š�Ñ r, éòÑ l, q;¤ÆVuà�òAí, Û“øz‘„ •éò‘Ç, Í(øÆVÞÚÇÿAø_J l Ñš�, �¶ÅÑCÅím$7, àÇ1, Ä òÆVí�¶ÅÑ 2πr, ]ÚÇ(5m$C ÅÑ 2πr, Ĥm$Þ� = 1 2 l(2πr) = πrl, ¤¹ÑòÆV5VÞ�, J,íj¶'1, 1 Êø�ñíÇ$�ÇA�ÞíÇ$, ]wÑ uý? Ç 1 wŸBbVõõu´æÊø_m$…í ¶Å�k…íÞ� (Ns6�¾ó�), }& à-: qm$íš�Ñ r, 2-iÑ θ (C �), †CÅ = rθ, ¶Å = 2r + rθ, Þ� = 1 2 r2θ, Yæ<� 2r + rθ = 1 2 r2θ, si ¾  r, j) θ = 4 r−2 (r �= 2), Ĥ)ƒ -�!�: càø_m$íš� �= 2, †ç 1 2 bçfÈ ��»û‚ ¬84�12~ θ = 4 r−2 v, ¤m$í¶Å.�k¤m$í Þ�� wŸcàBb�ìm$í¶Å, †Sv Þ�|×? ¹I 2r + rθ = L (ÑìM), † â AM ≥ GM ): L = 2r + rθ ≥ 2 √ 2r2θ ¹ 2r2θ ≤ L 2 4 FJ 1 2 r2θ ≤ L 2 16 Ñ|×Þ�� ¤v 2r = rθ FJθ = 2 (C�) ¹¶Å� ìÑ L v, †ç2-i = 2 (C�) v, ¤m $�|×Þ� L 2 16 � ¥5, �ìm$Þ�Ñ A, ¹q 1 2 r2θ = A ( A ÑìM)� † r2θ = 2A, FJ2r + rθ ≥ 2 √ 2r2θ = 2 √ 4A = 4 √ A Ñ|ü¶Å, ¤v 2r = rθ, ¹ θ = 2� 6ÿuz: Þ��ìÑ A v, †ç2-i = 2 (C�) v, ¤m$�|ü¶Å 4 √ A� (¤!�?ªâ L 2 16 = A, j) L = 4 √ A) #]ø_*3: *31. t„�¶Åím$Dä$,s6 í|×Þ�ó�� (key: q¶ÅÑ L, †|× Þ�ÌÑ L 2 16 )� -ÞBbT|ø©��Ém$í½æ, ¥<æñâ»p¿, ı ]?*2ä}w2 j¶51, �< G#] ;;_2−Ü: æ1: -Ç25m$ AOB 2, � AOB = 90◦, š� OA = OB = 1, }�J OA, OB Ñò�Êm$q¶TšÆ, °é(¶} íÞ�� Ç 2 j: 1. ©Q AC D BC † AC D BC u(, / $ AC D CB� s6Þ�í¸�k å$ 0C 5Þ� (ÑS?) 2. FJé(¶}Þ�= $ ACB 5Þ� = m$A0B 5Þ�−�AOB 5Þ� = 1 4 π×12−1 2 ×12= π 4 −1 2 (�jÀP)� æ2: £j$5iÅÑ1, }�J£j$íÝ õÑÆ-, 1Ñš�å4_ÆC, ó>à-Ç Fý, °é(¶MíÞ�� Ç 3 2çbçØ}øøm$51 3 j: ¥_½æíj¶'Ö, Bb� Œk“ìýìÜ”Vj}ªœjZ, l�ÜS ‚“ìýìÜ”? ÿu: � 2, Løií�j �kwFsií�j¸Á siDHiìý ��ísI� 1. I£j$ûÝõÑ ABCD, é(¶}í ûÝõÑ PQRS, (àÇ4Fý) Ç 4 2. ©Q PQ, QR, RS, SP D BP , BQ, † � PBQ = π 6 , (ÑS?)£j$ PQRS íÞ� = PQ 2 , YìýìÜ) PQ 2 = BP 2 +BQ 2 − 2BP × BQ × cos ( � PBQ) = 12 + 12 − 2× 1× 1× cos π 6 = 2− √ 3 3. $ PQ íÞ� = m$BPQ íÞ�−�BPQíÞ� = 1 2 × 12 × π 6 − 1 2 × 12 × sin π 6 = π 12 − 1 4 4. kuF°Þ� = £j$PQRSíÞ�+ 4_ $ PQ íÞ� = (2− √ 3) + 4( π 12 − 1 4 ) = π 3 + 1− √ 3(�jÀP) æ3: àÇ5m$ OAB 5š� r, Æ-iÑ 60◦, t°wqQä$ PQRS 5|×Þ�� Ç 5 j: 1. T � AOB í�}(> QR k M , † � MOA = 30◦, I � MOQ = θ, † OM = r cos θ, QM = r sin θ, FJQR=2r sin θ� PQ = r cos θ − r sin θ cot 30◦ = r(cos θ − sin θ √ 3) = 2r(sin 30◦ cos θ − cos 30◦ sin θ) = 2r sin (30◦ − θ)� 2. Þ� = 4r2 sin θ · sin (30◦ − θ) = 2r2[cos (2θ − 30◦)− cos 30◦] (“�A Ï)� 3. éÍ 2θ − 30◦ = 0◦ v�|×Þ� 2r2(1− √ 3 2 ) = (2−√3)r2� 4 bçfÈ ��»û‚ ¬84�12~ øJ,½æØ׃øO8$: #ìøm$, š�Ñ r, 2-iÑ 2α (r > 0, 0 < α < π 2 ), °¤m$qQä$ PQRS í|×Þ� (Q, R ÊÆC, P, S }Êsš �,)(¡©Ç5) I¥|×Þ�um$Þ�í S(α) I, t„ 1 2 < S(α) < 2 π � (70�rÅb¬�ø¼¨t æ) j: 1. |×Þ�",q°)Ñ r2 tan α 2 � 2. S(α) = r2 tan α 2 r2 2 (2α) = tan α 2 α = 1 2 tan α 2 α 2 , ÄÑ0 < α 2 < π 4 , FJ tan α 2 α 2 > 1 (¡© (ÞÇ9F„p54”), ] S(α) > 1 2 � 3. wŸç θ ∈ (0, π 4 ) v, J?„p tan θ θ ] Ó†ç θ = π 4 v, ÿª)ƒ S(α) í, Ì 2 π � àÇ6Fý Ç 6 �OAE m$ODG > m$OCE m$ODG = m$OEF m$OGB > �OEB m$OGB FJ �OAE m$ODG > �OEB m$OGB � ‚à a b > c d ⇒ a+c b+d > c d ª) �OAE+�OEB m$ODG+ m$OGB > �OEB m$OGB ¹ �OAB m$ODB > �OEB m$OGB FJ tan θ θ > tan φ φ , ku): θ > φ ⇒ tan θ θ > tanφ φ 4. ÄÑ0 < α 2 < π 4 FJS(α) = 1 2 tan α 2 α 2 < 1 2 tan π 4 π 4 = 2 π (ÇÕ°¬Ü�bçí°ç? ªàúiƒb�}íj¶„p f ′(θ) > 0, ku f(θ) ]Ó, ¤T f(θ) = tan θ θ )� æ4: �øm$ AOB w2-i � AOB Ñ θ, š�Ñ r, q P ÑC ÂB ,Løõ, A P ²sš� OA £ OB ®T�(, �—} �Ñ Q £ R, t„(¨ QR íÅÑøìM (Ç7)(82�rÅb¬�ù¼¨tæ)� Ç 7 „p: 1. ©Q OP q � AOP = α, � BOP = β, † α + β = θ, / � QPR = π − θ, Ê Rt�OPQ 2, PQ = r sinα, ¢Ê Rt�OPR 2, PR = r sin β� 2. Ê �PQR 2, â ì ý ì Ü ) QR 2 = PQ 2 + PR 2 − 2PQ PR cos( � QPR) = r2 sin2 α + r2 sin2 β 2çbçØ}øøm$51 5 −2r2 sinα sin β cos (π − θ) = r2(sin2 α+sin2 β+2 sinα sin β cos θ) = r2( 1− cos 2α 2 + 1− cos 2β 2 +2 sinα sin β cos θ) = r2[1− 1 2 (cos 2α+ cos 2β) +2 sinα sin β cos θ] = r2[1− 1 2 · 2 cos(α + β) cos(α− β) +2 sinα sin β cos θ] = r2[1−cos θ cos(α−β)+(cos(α−β) − cos(α + β)) cos θ] = r2[1− cos θ cos(α− β) + cos(α− β) cos θ − cos2 θ] = r2(1− cos2θ) = r2 sin2 θ Ñ ì M *32: Êm$ AOB 2 O Ñ2-, OA = OB = r, P ÑÆC ÂB ,Lø õ, 7 P B OA í�×Ñ a, B OB í� ×Ñ b, tø r J a, b [ý5 (77×ç:, : 2√ 3 √ a2+ab+b2)� *33: Ç82, m$ AOB 5š�Ñ 14, 2-iÑ θ, Ê ÂB ,�øõ P , â P ú OA T�ò(¨ PQ, wÅ13, P ú OB T�ò(¨ PR, wÅ11, ° (1) θ (2) é( ¶}íÞ� ( : (1) 2π 3 (2) 196π 3 −47√3)� Ç 8 w Ÿ B b V } ø _ ½ æ, ] � ) ú i ƒ b 2, sin x í û ƒ b Ñcos x, cos x í û ƒ b Ñ − sin x ý? ¹ d sin x dx = cos x, d cos x dx = − sin x, ¥ s _ t � í û | �“m $”6 ˆ , É [: Ÿ V B b ª � Œ k“m $”l û | - � 4 ” (1), y â (1) û | (2) 7 | ( ª û | sin x, cos x í � } t �: (1) q −π 2 < x < π 2 , x �= 0, † | sinx| < |x| < | tanx|� (2) lim x→0 sinx x = 1� (1) „p: } - � s � 8 $ ª W: 1. 0 < x < π 2 : à - Ç 9À P Æ 2, � AOB = x, ¬ A T Æ í ~ ( > −−→ OB k C, 6 bçfÈ ��»û‚ ¬84�12~ Ç 9 † �AOB < m $AOB < �AOC, â m $ Þ � = 1 2 r2θ (r [ š �, θ [ 2 - i J C � l), ª ) 1 2 sin x < 1 2 x < 1 2 tan x, F J sin x < x < tan x, ¹ | sinx| < |x| < | tanx|� 2. −π 2 < x < 0: ¹ 0 < −x < π 2 , â 1ª ) sin(−x) < −x < tan(−x), −sin x < −x < −tan x ¹ | sin x| < |x| < | tanx|� ã , 1.2ø: −π 2 < x < π 2 , x �= 0, 0 � | sinx| < |x| < | tanx|� (2) „p: F Jx→ 0 F J− π 2 < x < π 2 , x �= 0 â (1) ø | sin x| < |x| < | tanx| = | sinx| | cos x| ⇒ 1 < |x|| sinx| < 1| cos x| ⇒ 1 < xsinx < 1cos x (Ä Ñ sin x D x ° U, 7 cosx > 0) ⇒ cosx < sinx x < 1 Ä Ñx → 0 v, cosx → 1 F J â ˜ Ñ ì Ü ø lim x→0 sinx x = 1� k u d sinx dx ∣∣∣∣∣ x=a = lim x→a sin x− sin a x− a = lim x→a 2 cos x+a 2 · sin x−a 2 x− a (} ä ‚ à sin x − sin y = 2 cos x+ y 2 · sin x− y 2 ) = lim x→a cos x+ a 2 · sin x−a 2 x−a 2 = lim x→a cos x+ a 2 · lim x−a 2 →0 sin x−a 2 x−a 2 = cos a · 1 = cos a F J d sin x dx = cos x, ° Ü ª ) d cosx dx = − sin x | ( B b y V „ p ø _ ½ æ: 7 Þ , s õ È 5 C Å J ¦ ¬ v s õ 5 × Æ í Æ C Ñ | s� „p: 1. q 7 Þ S : x2 + y2 + z2 = r2 , s õ A� B (q ° ï �) (ª J ø 7 _ ç í T � � 7 ® ¤ b °, ] . Ü w ø O 4)� 2. q A� B s õ F Ê í ì ï � Ñ φ (φ = 90◦−φ′, φ′ [ ï �, ” ï ¦+, œ ï ¦−)(¡ © Ç 10) I � AOB = θ, ü Æ C 5 Æ - Ñ E, � AEB = α (0 < θ, α < π), × Æ C ÂB Å Ñ S, ü Æ C ÂB Å Ñ S ′ (Ç 11), † ü Æ C 5 š � EB = r · sinφ (Ç 12)� S = rθ, S ′ = EB · α = (r sin φ)α = rα sinφ� 2çbçØ}øøm$51 7 3. ø �EAB ÷ AB � � U D �OAB u � Þ (Ç 13), † � BOE = θ 2 , � BEO = α 2 � â £ ý ì Ü ) OB sin α 2 = BE sin θ 2 , ¤ 2 OB = r, BE = r sin φ� F Jr sin θ 2 = r sinφ sin α 2 , ¹ sin θ 2 = sinφ sin α 2 , F J sinφ 1 = sin θ 2 sin α 2 (∗)� (φ′ [ P õ í ï �, θ [ P õ í % ��) Ç 10 Ç 11 Ç 12 Ç 13 Ûk„p S ′ > S ¹k„ S ′ S = rα sinφ rθ = α sinφ θ > 1, ¹k„ θ α < sinφ 1 â, Þ (∗) ø: ¹k„ θ2α 2 < sin θ 2 sin α 2 , 6ÿu sinα/2 α/2 < sin θ/2 θ/2 , Ä×Æš� OB > ü Æš� EB, ] α/2 > θ/2� FJ Éb „p f(x) = sinx x ÑÁƒb¹ª, (w2 0 < x < π 2 )� 4. @àúi5¸it�q 0 < β < α < π 2 I α = β + r, r > 0, † sin β β − sinα α = α sin β − β sinα αβ 8 bçfÈ ��»û‚ ¬84�12~ = 1 αβ [(β + r) sin β − β sin(β + r)] = 1 αβ [(β + r) sin β − β(sin β cos r +cosβ sin r)] = 1 αβ [β sin β(1− cos r) + r sin β −β cosβ sin r] 7 1− cos r > 0, / r sin β − β cos β sin r = r sin β − (β cosβ) sin r > r sin β − sin β sin r (ÄÑβ < tan β = sin β cos β ) = sin β(r − sin r) > 0 (ÄÑr > sin r) FJ sin β β > sinα α , çβ < α v, ] f(x) = sinx x ÑÁƒb� FJ…½æ)„� ¥ ÿ u B b � ì œ * Ÿ � ¸ % Ç Í ì � « ”, ì œ . ò Q � ² Ç Í ì, 7 b ÷ ƒ ” ” œ j ¯ G C ˇ ¿ , ˛ ì W í − Ü, Ä Ñ ( 6 í N ( u Ê ø × Æ C , í í ]� ¡5’e 1. «×bçÍ3): 2M¬Å�çn?Ü� ²‰º�ò2bç¬ttæ£ }&ùÕ, «”�2Md“+E«�RWãº}� 2. Å�«É�×Ë23): ò2bçkõ`‡ �ú4, «”�`>¶2�`>−|�� —…dT6L`kô�Á2Þäò2—