解直角三角形（三角函数问题）
\u89e3\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u4e09\u89d2\u51fd\u6570\u503c\u89e3\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u51fd\u6570\u503c\u5217\u4e3e\uff1a
sin1=0.01745240643728351 sin2=0.03489949670250097 sin3=0.05233595624294383
sin4=0.0697564737441253 sin5=0.08715574274765816 sin6=0.10452846326765346
sin7=0.12186934340514747 sin8=0.13917310096006544 sin9=0.15643446504023087
sin10=0.17364817766693033 sin11=0.1908089953765448 sin12=0.20791169081775931
sin13=0.22495105434386497 sin14=0.24192189559966773 sin15=0.25881904510252074
sin16=0.27563735581699916 sin17=0.2923717047227367 sin18=0.3090169943749474
sin19=0.3255681544571567 sin20=0.3420201433256687 sin21=0.35836794954530027
sin22=0.374606593415912 sin23=0.3907311284892737 sin24=0.40673664307580015
sin25=0.42261826174069944 sin26=0.4383711467890774 sin27=0.45399049973954675
sin28=0.4694715627858908 sin29=0.48480962024633706 sin30=0.49999999999999994
sin31=0.5150380749100542 sin32=0.5299192642332049 sin33=0.544639035015027
sin34=0.5591929034707468 sin35=0.573576436351046 sin36=0.5877852522924731
sin37=0.6018150231520483 sin38=0.6156614753256583 sin39=0.6293203910498375
sin40=0.6427876096865392 sin41=0.6560590289905073 sin42=0.6691306063588582
sin43=0.6819983600624985 sin44=0.6946583704589972 sin45=0.7071067811865475
sin46=0.7193398003386511 sin47=0.7313537016191705 sin48=0.7431448254773941
sin49=0.7547095802227719 sin50=0.766044443118978 sin51=0.7771459614569708
sin52=0.7880107536067219 sin53=0.7986355100472928 sin54=0.8090169943749474
sin55=0.8191520442889918 sin56=0.8290375725550417 sin57=0.8386705679454239
sin58=0.848048096156426 sin59=0.8571673007021122 sin60=0.8660254037844386
sin61=0.8746197071393957 sin62=0.8829475928589269 sin63=0.8910065241883678
sin64=0.898794046299167 sin65=0.9063077870366499 sin66=0.9135454576426009
sin67=0.9205048534524404 sin68=0.9271838545667873 sin69=0.9335804264972017
sin70=0.9396926207859083 sin71=0.9455185755993167 sin72=0.9510565162951535
sin73=0.9563047559630354 sin74=0.9612616959383189 sin75=0.9659258262890683
sin76=0.9702957262759965 sin77=0.9743700647852352 sin78=0.9781476007338057
sin79=0.981627183447664 sin80=0.984807753012208 sin81=0.9876883405951378
sin82=0.9902680687415704 sin83=0.992546151641322 sin84=0.9945218953682733
sin85=0.9961946980917455 sin86=0.9975640502598242 sin87=0.9986295347545738
sin88=0.9993908270190958 sin89=0.9998476951563913
sin90=1
cos1=0.9998476951563913 cos2=0.9993908270190958 cos3=0.9986295347545738
cos4=0.9975640502598242 cos5=0.9961946980917455 cos6=0.9945218953682733
cos7=0.992546151641322 cos8=0.9902680687415704 cos9=0.9876883405951378
cos10=0.984807753012208 cos11=0.981627183447664 cos12=0.9781476007338057
cos13=0.9743700647852352 cos14=0.9702957262759965 cos15=0.9659258262890683
cos16=0.9612616959383189 cos17=0.9563047559630355 cos18=0.9510565162951535
cos19=0.9455185755993168 cos20=0.9396926207859084 cos21=0.9335804264972017
cos22=0.9271838545667874 cos23=0.9205048534524404 cos24=0.9135454576426009
cos25=0.9063077870366499 cos26=0.898794046299167 cos27=0.8910065241883679
cos28=0.882947592858927 cos29=0.8746197071393957 cos30=0.8660254037844387
cos31=0.8571673007021123 cos32=0.848048096156426 cos33=0.838670567945424
cos34=0.8290375725550417 cos35=0.8191520442889918 cos36=0.8090169943749474
cos37=0.7986355100472928 cos38=0.7880107536067219 cos39=0.7771459614569709
cos40=0.766044443118978 cos41=0.754709580222772 cos42=0.7431448254773942
cos43=0.7313537016191705 cos44=0.7193398003386512 cos45=0.7071067811865476
cos46=0.6946583704589974 cos47=0.6819983600624985 cos48=0.6691306063588582
cos49=0.6560590289905074 cos50=0.6427876096865394 cos51=0.6293203910498375
cos52=0.6156614753256583 cos53=0.6018150231520484 cos54=0.5877852522924731
cos55=0.5735764363510462 cos56=0.5591929034707468 cos57=0.5446390350150272
cos58=0.5299192642332049 cos59=0.5150380749100544 cos60=0.5000000000000001
cos61=0.4848096202463371 cos62=0.46947156278589086 cos63=0.4539904997395468
cos64=0.43837114678907746 cos65=0.42261826174069944 cos66=0.4067366430758004
cos67=0.3907311284892737 cos68=0.3746065934159122 cos69=0.35836794954530015
cos70=0.3420201433256688 cos71=0.32556815445715675 cos72=0.30901699437494745
cos73=0.29237170472273677 cos74=0.27563735581699916 cos75=0.25881904510252074
cos76=0.24192189559966767 cos77=0.22495105434386514 cos78=0.20791169081775923
cos79=0.19080899537654491 cos80=0.17364817766693041 cos81=0.15643446504023092
cos82=0.13917310096006546 cos83=0.12186934340514749 cos84=0.10452846326765346
cos85=0.08715574274765836 cos86=0.06975647374412523 cos87=0.052335956242943966
cos88=0.03489949670250108 cos89=0.0174524064372836
cos90=0
tan1=0.017455064928217585 tan2=0.03492076949174773 tan3=0.052407779283041196
tan4=0.06992681194351041 tan5=0.08748866352592401 tan6=0.10510423526567646
tan7=0.1227845609029046 tan8=0.14054083470239145 tan9=0.15838444032453627
tan10=0.17632698070846497 tan11=0.19438030913771848 tan12=0.2125565616700221
tan13=0.2308681911255631 tan14=0.24932800284318068 tan15=0.2679491924311227
tan16=0.2867453857588079 tan17=0.30573068145866033 tan18=0.3249196962329063
tan19=0.34432761328966527 tan20=0.36397023426620234 tan21=0.3838640350354158
tan22=0.4040262258351568 tan23=0.4244748162096047 tan24=0.4452286853085361
tan25=0.4663076581549986 tan26=0.4877325885658614 tan27=0.5095254494944288
tan28=0.5317094316614788 tan29=0.554309051452769 tan30=0.5773502691896257
tan31=0.6008606190275604 tan32=0.6248693519093275 tan33=0.6494075931975104
tan34=0.6745085168424265 tan35=0.7002075382097097 tan36=0.7265425280053609
tan37=0.7535540501027942 tan38=0.7812856265067174 tan39=0.8097840331950072
tan40=0.8390996311772799 tan41=0.8692867378162267 tan42=0.9004040442978399
tan43=0.9325150861376618 tan44=0.9656887748070739 tan45=0.9999999999999999
tan46=1.0355303137905693 tan47=1.0723687100246826 tan48=1.1106125148291927
tan49=1.1503684072210092 tan50=1.19175359259421 tan51=1.234897156535051
tan52=1.2799416321930785 tan53=1.3270448216204098 tan54=1.3763819204711733
tan55=1.4281480067421144 tan56=1.4825609685127403 tan57=1.5398649638145827
tan58=1.6003345290410506 tan59=1.6642794823505173 tan60=1.7320508075688767
tan61=1.8040477552714235 tan62=1.8807264653463318 tan63=1.9626105055051503
tan64=2.050303841579296 tan65=2.1445069205095586 tan66=2.246036773904215
tan67=2.355852365823753 tan68=2.4750868534162946 tan69=2.6050890646938023
tan70=2.7474774194546216 tan71=2.904210877675822 tan72=3.0776835371752526
tan73=3.2708526184841404 tan74=3.4874144438409087 tan75=3.7320508075688776
tan76=4.0107809335358455 tan77=4.331475874284153 tan78=4.704630109478456
tan79=5.144554015970307 tan80=5.671281819617707 tan81=6.313751514675041
tan82=7.115369722384207 tan83=8.144346427974593 tan84=9.514364454222587
tan85=11.43005230276132 tan86=14.300666256711942 tan87=19.08113668772816
tan88=28.636253282915515 tan89=57.289961630759144
tan90=(\u65e0\u9650)
\u5728\u76f4\u89d2\u4e09\u89d2\u5f62\u4e2d\uff0c\u9664\u76f4\u89d2\u5916\uff0c\u4e00\u5171\u6709\u4e94\u4e2a\u5143\u7d20\uff0c\u5373\u4e09\u6761\u8fb9\u548c\u4e24\u4e2a\u9510\u89d2\uff0c\u7531\u76f4\u89d2\u4e09\u89d2\u5f62\u4e2d\u9664\u76f4\u89d2\u5916\u7684\u5df2\u77e5\u5143\u7d20\u6c42\u51fa\u6240\u6709\u672a\u77e5\u5143\u7d20\u7684\u8fc7\u7a0b\u53eb\u505a\u89e3\u76f4\u89d2\u4e09\u89d2\u5f62\u3002
\u5728\u89e3\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9996\u5148\u8981\u660e\u786e\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u7684\u5b9a\u4e49\uff1a
\u5f0f\u4e2d\u5927\u5199\u5b57\u6bcd\u8868\u793a\u89d2\u5ea6\uff0c\u5c0f\u5199\u5b57\u6bcd\u8868\u793a\u4e09\u89d2\u5f62\u8fb9\u957f\uff0cc\u8868\u793a\u76f4\u89d2\u8fb9\u957f\u3002
\u53ef\u89c1\uff0c\u6bcf\u4e2a\u5f0f\u5b50\u4e2d\u542b\u6709\u4e09\u89d2\u5f62\u76843\u4e2a\u5143\u7d20\uff0c\u5f533\u8005\u4e4b\u4e2d\u67092\u8005\u5df2\u77e5\u65f6\uff0c\u5b83\u5c31\u8f6c\u5316\u4e3a\u4e00\u4e2a\u4e00\u5143\u65b9\u7a0b\uff0c\u89e3\u65b9\u7a0b\u5373\u53ef\u6c42\u51fa\u672a\u77e5\u5143\u7d20\u3002
\u6839\u636e\u5df2\u77e5\u5143\u7d20\u79cd\u7c7b\u7684\u4e0d\u540c\uff0c\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u89e3\u6cd5\u53ef\u4ee5\u5f52\u7eb3\u4e3a\u4ee5\u4e0b4\u7c7b\uff0c\u5982\u8868\u6240\u793a\uff1a
由正弦定理，有sinA/BC=sinB/AC
10/sin(3x)=6/sinx
10/(3sinx-4sin³x)=6/sinx
10/(3-4sin²x)=6
sin²x=1/3
sinx=√3/3
cosx=√6/3
sin(2x)=2sinxcosx=2√2/3
cos(2x)=1-2sin²x=1/3
sin(4x)=2sin(2x)cos(2x)=4√2/9
sin(п-4x)=sinx=4√2/9
sinC=4√2/9
给一个简单的解法:
绛旓細鍒濅腑瀛︿範鐨勯攼瑙涓夎鍑芥暟鍊肩殑瀹氫箟鏂规硶鏄湪鐩磋涓夎褰涓畾涔夌殑锛屽嵆鎶婅繖涓鏀惧埌濡傚浘鎵绀虹殑鐩磋涓夎褰腑锛屽幓姹傝В銆傚叾浠栨柟娉曪細1銆佹寮﹀畾鐞嗘眰瑙ｏ紝鍗砤/sinA = b/sinB =c/sinC = 2r=D 2銆佽繍鐢ㄤ綑寮﹀畾鐞嗭紝鐩存帴杩愮敤瀹冨彲瑙ｅ喅涓绫诲凡鐭ヤ笁瑙掑舰涓よ竟鍙婂す瑙掓眰绗笁杈规垨鑰呮槸宸茬煡涓変釜杈规眰涓夎鐨勯棶棰橈紱鍗砪os A=(b&#...
绛旓細鐢诲嚭鐩磋涓夎琛岋紙306090搴︼級30搴︽墍瀵圭殑鐩磋杈逛负鏂滆竟鐨勪竴鍗婏紝鏍规嵁鍕捐偂瀹氱悊鍙亣璁句笁杈逛负12鏍瑰彿3鍐嶆牴鎹搴﹀氨鑳界煡閬涓夎鍑芥暟浜嗭綖sin60==鈭3/2
绛旓細1銆鐩磋涓夎褰涓殑鍩烘湰棰樺瀷锛氬凡鐭ヤ袱杈癸紝姹傞攼瑙掞細鐩存帴杩愮敤涓夎鍑芥暟姒傚康锛涘凡鐭ヤ竴杈瑰拰閿愯锛屽垎涓夌锛氱粰閭昏竟銆佸杈广佹枩杈癸紝鍒嗗埆姹傚嚭鍙﹀涓よ竟銆2銆佷袱涓洿瑙掍笁瑙掑舰缁勫悎鍥惧舰涓殑鍩烘湰棰樺瀷锛氫竴鑸兘缁欎袱涓攼瑙掞紙涓嶅悓涓夎褰腑锛夛紝鍐嶇粰涓鏉¤竟锛岃屾眰鍏跺畠杈广備竴缁勭洿瑙掕竟閲嶅悎锛涗竴鏉＄洿瑙掕竟鍜屽彟涓涓夎褰㈢殑鏂滆竟閲嶅悎锛涚殑...
绛旓細璁続=3x锛孊=x锛宑=锌-4x 鐢辨寮﹀畾鐞嗭紝鏈塻inA/BC=sinB/AC 10/sin(3x)=6/sinx 10/(3sinx-4sin³x)=6/sinx 10/(3-4sin²x)=6 sin²x=1/3 sinx=鈭3/3 cosx=鈭6/3 sin(2x)=2sinxcosx=2鈭2/3 cos(2x)=1-2sin²x=1/3 sin(4x)=2sin(2x)cos(2x)=...
绛旓細鈭礎B鈭F锛屸埓BM=BC脳sin30掳=10脳=5锛孋M=BC脳cos30掳=15锛屽湪鈻矱FD涓紝鈭燜=90掳锛屸垹E=45掳锛屸埓鈭燛DF=45掳锛屸埓MD=BM=5锛屸埓CD=CM-MD=15-5锛庛愮偣璇勩戞湰棰樿冩煡浜嗚В鐩磋涓夎褰鐨勬ц川鍙婂钩琛岀嚎鐨勬ц川锛岄毦搴﹁緝澶э紝瑙ｇ瓟姝ょ被棰樼洰鐨勫叧閿牴鎹鎰忓缓绔嬩笁瑙掑舰鍒╃敤鎵瀛︾殑涓夎鍑芥暟鐨勫叧绯昏繘琛岃В绛旓紟
绛旓細杩欎簺鍏紡灏辫兘杞崲涓轰竴鍏冩柟绋嬶紝閫氳繃瑙ｆ柟绋嬶紝鎴戜滑灏辫兘鎵惧埌鍓╀綑鐨勬湭鐭ュ厓绱犮傛崲鍙ヨ瘽璇达紝瑙ｇ洿瑙掍笁瑙掑舰鐨勮繃绋嬪氨鏄繍鐢涓夎鍑芥暟鐨勫叧绯伙紝灏嗗凡鐭ョ殑涓や釜杈归暱鎴栬搴︼紝杞寲涓哄崟涓鏈煡鏁扮殑鏂圭▼锛屼粠鑰岀簿纭湴鎵惧嚭涓夎褰㈢殑鍏朵粬灏哄鎴栬搴︺傝繖绉嶆柟娉曞湪娴嬮噺銆佸嚑浣曢棶棰樺拰鐗╃悊瀛︿腑閮芥瀬涓哄父瑙侊紝鏄悊瑙ｅ拰搴旂敤涓夎瀛︾殑鍩虹銆
绛旓細涓夎鍑芥暟涓鍏辨湁6涓:鐩磋涓夎褰涓:姝ｅ鸡:sin 瀵硅竟姣旀枩杈 浣欏鸡:cos 閭昏竟姣旀枩杈 姝ｅ垏:tan 瀵硅竟姣旈偦杈 浣欏垏:cot 閭昏竟姣斿杈 姝ｅ壊:csc 鏂滆竟姣斿杈 浣欏壊:sec 鏂滆竟姣旈偦杈 璁句笁瑙掑舰涓変釜鍐呰鍒嗗埆涓篈,B,C;瀵硅竟鍒嗗埆涓篴,b,c 姝ｅ鸡瀹氱悊:a/sinA=b/sinB=c/sinC=2R,(R涓鸿涓夎褰㈠鎺ュ渾鍗婂緞)浣欏鸡瀹氱悊:c&...
绛旓細鐢辩洿瑙掍笁瑙掑舰涓凡鐭ョ殑杈瑰拰瑙掞紝璁＄畻鍑烘湭鐭ョ殑杈瑰拰瑙掔殑杩囩▼锛屽彨鍋瑙ｇ洿瑙掍笁瑙掑舰.閿愯伪0掳30掳45掳60掳90掳sin 伪锛堟寮︼級01/2鈭2/2鈭3/21cos伪锛堜綑寮︼級1鈭3/2鈭2/21/20tan 伪锛堟鍒囷級0鈭3/31鈭3涓嶅瓨鍦╟ot 伪锛堜綑鍒囷級涓嶅瓨鍦ㄢ垰31鈭3/30 閿愯涓夎鍑芥暟 sinA=a/c,cosA=b/c,tanA=a/b,...
绛旓細鍦ㄧ洿瑙掍笁瑙掑舰涓紝闄ょ洿瑙掑锛屼竴鍏辨湁浜斾釜鍏冪礌锛屽嵆涓夋潯杈瑰拰涓や釜閿愯锛岀敱鐩磋涓夎褰腑闄ょ洿瑙掑鐨勫凡鐭ュ厓绱犳眰鍑烘墍鏈夋湭鐭ュ厓绱犵殑杩囩▼鍙仛瑙ｇ洿瑙掍笁瑙掑舰銆傚湪瑙ｇ洿瑙掍笁瑙掑舰鐨勮繃绋嬩腑锛岄鍏堣鏄庣‘閿愯涓夎鍑芥暟鐨勫畾涔夛細寮忎腑澶у啓瀛楁瘝琛ㄧず瑙掑害锛屽皬鍐欏瓧姣嶈〃绀轰笁瑙掑舰杈归暱锛宑琛ㄧず鐩磋杈归暱銆傚彲瑙侊紝姣忎釜寮忓瓙涓惈鏈変笁瑙掑舰鐨3涓...
绛旓細鍦ㄧ洿瑙掍笁瑙掑舰涓紝闄ょ洿瑙掑锛屼竴鍏辨湁浜斾釜鍏冪礌锛屽嵆涓夋潯杈瑰拰涓や釜閿愯锛岀敱鐩磋涓夎褰腑闄ょ洿瑙掑鐨勫凡鐭ュ厓绱犳眰鍑烘墍鏈夋湭鐭ュ厓绱犵殑杩囩▼鍙仛瑙ｇ洿瑙掍笁瑙掑舰銆傚湪瑙ｇ洿瑙掍笁瑙掑舰鐨勮繃绋嬩腑锛岄鍏堣鏄庣‘閿愯涓夎鍑芥暟鐨勫畾涔夛細寮忎腑澶у啓瀛楁瘝琛ㄧず瑙掑害锛屽皬鍐欏瓧姣嶈〃绀轰笁瑙掑舰杈归暱锛宑琛ㄧず鐩磋杈归暱銆傚彲瑙侊紝姣忎釜寮忓瓙涓惈鏈変笁瑙掑舰鐨3涓...
