如图的一道初三锐角三角函数题 求大神教 学霸来T^T 一道锐角三角函数 题（要有过程）

\u521d\u4e09\u4e0b\u518c\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u6240\u6709\u7ec3\u4e60\u9898\u7684\u7b54\u6848\uff1f

\u4e09\u89d2\u51fd\u6570\u662f\u6570\u5b66\u4e2d\u5c5e\u4e8e\u521d\u7b49\u51fd\u6570\u4e2d\u7684\u8d85\u8d8a\u51fd\u6570\u7684\u4e00\u7c7b\u51fd\u6570\u3002\u5b83\u4eec\u7684\u672c\u8d28\u662f\u4efb\u610f\u89d2\u7684\u96c6\u5408\u4e0e\u4e00\u4e2a\u6bd4\u503c\u7684\u96c6\u5408\u7684\u53d8\u91cf\u4e4b\u95f4\u7684\u6620\u5c04\u3002\u901a\u5e38\u7684\u4e09\u89d2\u51fd\u6570\u662f\u5728\u5e73\u9762\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d\u5b9a\u4e49\u7684\uff0c\u5176\u5b9a\u4e49\u57df\u4e3a\u6574\u4e2a\u5b9e\u6570\u57df\u3002\u53e6\u4e00\u79cd\u5b9a\u4e49\u662f\u5728\u76f4\u89d2\u4e09\u89d2\u5f62\u4e2d\uff0c\u4f46\u5e76\u4e0d\u5b8c\u5168\u3002\u73b0\u4ee3\u6570\u5b66\u628a\u5b83\u4eec\u63cf\u8ff0\u6210\u65e0\u7a77\u6570\u5217\u7684\u6781\u9650\u548c\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\uff0c\u5c06\u5176\u5b9a\u4e49\u6269\u5c55\u5230\u590d\u6570\u7cfb\u3002
\u3000\u3000\u7531\u4e8e\u4e09\u89d2\u51fd\u6570\u7684\u5468\u671f\u6027\uff0c\u5b83\u5e76\u4e0d\u5177\u6709\u5355\u503c\u51fd\u6570\u610f\u4e49\u4e0a\u7684\u53cd\u51fd\u6570\u3002
\u3000\u3000\u4e09\u89d2\u51fd\u6570\u5728\u590d\u6570\u4e2d\u6709\u8f83\u4e3a\u91cd\u8981\u7684\u5e94\u7528\u3002\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u4e09\u89d2\u51fd\u6570\u4e5f\u662f\u5e38\u7528\u7684\u5de5\u5177\u3002
\u3000\u3000\u57fa\u672c\u521d\u7b49\u5185\u5bb9
\u3000\u3000\u5b83\u6709\u516d\u79cd\u57fa\u672c\u51fd\u6570(\u521d\u7b49\u57fa\u672c\u8868\u793a)\uff1a
\u3000\u3000\u51fd\u6570\u540d \u6b63\u5f26 \u4f59\u5f26 \u6b63\u5207 \u4f59\u5207 \u6b63\u5272 \u4f59\u5272
\u3000\u3000\u5728\u5e73\u9762\u76f4\u89d2\u5750\u6807\u7cfbxOy\u4e2d\uff0c\u4ece\u70b9O\u5f15\u51fa\u4e00\u6761\u5c04\u7ebfOP\uff0c\u8bbe\u65cb\u8f6c\u89d2\u4e3a\u03b8\uff0c\u8bbeOP=r\uff0cP\u70b9\u7684\u5750\u6807\u4e3a\uff08x\uff0cy\uff09\u6709
\u3000\u3000\u6b63\u5f26\u51fd\u6570 sin\u03b8=y/r \u8bfb\u4f5c\uff1a\u6492\u5e94
\u3000\u3000\u4f59\u5f26\u51fd\u6570 cos\u03b8=x/r \u8bfb\u4f5c\uff1a\u8003\u6492\u5e94
\u3000\u3000\u6b63\u5207\u51fd\u6570 tan\u03b8=y/x\u8bfb\u4f5c\uff1a\u575b\u9876\u8d34
\u3000\u3000\u4f59\u5207\u51fd\u6570 cot\u03b8=x/y\u8bfb\u4f5c\uff1a\u8003\u575b\u9876\u8d34
\u3000\u3000\u6b63\u5272\u51fd\u6570 sec\u03b8=r/x\u8bfb\u4f5c\uff1a\u585e\u6839\u57fa
\u3000\u3000\u4f59\u5272\u51fd\u6570 csc\u03b8=r/y\u8bfb\u4f5c\uff1a\u8003\u585e\u6839\u57fa
\u3000\u3000\uff08\u659c\u8fb9\u4e3ar\uff0c\u5bf9\u8fb9\u4e3ay\uff0c\u90bb\u8fb9\u4e3ax\u3002\uff09
\u3000\u3000\u4ee5\u53ca\u4e24\u4e2a\u4e0d\u5e38\u7528\uff0c\u5df2\u8d8b\u4e8e\u88ab\u6dd8\u6c70\u7684\u51fd\u6570\uff1a
\u3000\u3000\u6b63\u77e2\u51fd\u6570 versin\u03b8 =1-cos\u03b8
\u3000\u3000\u4f59\u77e2\u51fd\u6570 covers\u03b8 =1-sin\u03b8
[\u7f16\u8f91\u672c\u6bb5]\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u95f4\u7684\u57fa\u672c\u5173\u7cfb\u5f0f\uff1a
\u3000\u3000\u00b7\u5e73\u65b9\u5173\u7cfb\uff1a
\u3000\u3000sin^2(\u03b1)+cos^2(\u03b1)=1 cos^2a=(1+cos2a)/2
\u3000\u3000tan^2(\u03b1)+1=sec^2(\u03b1) sin^2a=(1-cos2a)/2
\u3000\u3000cot^2(\u03b1)+1=csc^2(\u03b1)
\u3000\u3000\u00b7\u79ef\u7684\u5173\u7cfb\uff1a
\u3000\u3000sin\u03b1=tan\u03b1*cos\u03b1
\u3000\u3000cos\u03b1=cot\u03b1*sin\u03b1
\u3000\u3000tan\u03b1=sin\u03b1*sec\u03b1
\u3000\u3000cot\u03b1=cos\u03b1*csc\u03b1
\u3000\u3000sec\u03b1=tan\u03b1*csc\u03b1
\u3000\u3000csc\u03b1=sec\u03b1*cot\u03b1
\u3000\u3000\u00b7\u5012\u6570\u5173\u7cfb\uff1a
\u3000\u3000tan\u03b1\u00b7cot\u03b1=1
\u3000\u3000sin\u03b1\u00b7csc\u03b1=1
\u3000\u3000cos\u03b1\u00b7sec\u03b1=1
\u3000\u3000\u76f4\u89d2\u4e09\u89d2\u5f62ABC\u4e2d,
\u3000\u3000\u89d2A\u7684\u6b63\u5f26\u503c\u5c31\u7b49\u4e8e\u89d2A\u7684\u5bf9\u8fb9\u6bd4\u659c\u8fb9,
\u3000\u3000\u4f59\u5f26\u7b49\u4e8e\u89d2A\u7684\u90bb\u8fb9\u6bd4\u659c\u8fb9
\u3000\u3000\u6b63\u5207\u7b49\u4e8e\u5bf9\u8fb9\u6bd4\u90bb\u8fb9,
\u3000\u3000\u00b7\u4e09\u89d2\u51fd\u6570\u6052\u7b49\u53d8\u5f62\u516c\u5f0f
\u3000\u3000\u00b7\u4e24\u89d2\u548c\u4e0e\u5dee\u7684\u4e09\u89d2\u51fd\u6570\uff1a
\u3000\u3000cos(\u03b1+\u03b2)=cos\u03b1\u00b7cos\u03b2-sin\u03b1\u00b7sin\u03b2
\u3000\u3000cos(\u03b1-\u03b2)=cos\u03b1\u00b7cos\u03b2+sin\u03b1\u00b7sin\u03b2
\u3000\u3000sin(\u03b1\u00b1\u03b2)=sin\u03b1\u00b7cos\u03b2\u00b1cos\u03b1\u00b7sin\u03b2
\u3000\u3000tan(\u03b1+\u03b2)=(tan\u03b1+tan\u03b2)/(1-tan\u03b1\u00b7tan\u03b2)
\u3000\u3000tan(\u03b1-\u03b2)=(tan\u03b1-tan\u03b2)/(1+tan\u03b1\u00b7tan\u03b2)
\u3000\u3000\u00b7\u4e09\u89d2\u548c\u7684\u4e09\u89d2\u51fd\u6570\uff1a
\u3000\u3000sin(\u03b1+\u03b2+\u03b3)=sin\u03b1\u00b7cos\u03b2\u00b7cos\u03b3+cos\u03b1\u00b7sin\u03b2\u00b7cos\u03b3+cos\u03b1\u00b7cos\u03b2\u00b7sin\u03b3-sin\u03b1\u00b7sin\u03b2\u00b7sin\u03b3
\u3000\u3000cos(\u03b1+\u03b2+\u03b3)=cos\u03b1\u00b7cos\u03b2\u00b7cos\u03b3-cos\u03b1\u00b7sin\u03b2\u00b7sin\u03b3-sin\u03b1\u00b7cos\u03b2\u00b7sin\u03b3-sin\u03b1\u00b7sin\u03b2\u00b7cos\u03b3
\u3000\u3000tan(\u03b1+\u03b2+\u03b3)=(tan\u03b1+tan\u03b2+tan\u03b3-tan\u03b1\u00b7tan\u03b2\u00b7tan\u03b3)/(1-tan\u03b1\u00b7tan\u03b2-tan\u03b2\u00b7tan\u03b3-tan\u03b3\u00b7tan\u03b1)
\u3000\u3000\u00b7\u8f85\u52a9\u89d2\u516c\u5f0f\uff1a
\u3000\u3000Asin\u03b1+Bcos\u03b1=(A^2+B^2)^(1/2)sin(\u03b1+t)\uff0c\u5176\u4e2d
\u3000\u3000sint=B/(A^2+B^2)^(1/2)
\u3000\u3000cost=A/(A^2+B^2)^(1/2)
\u3000\u3000tant=B/A
\u3000\u3000Asin\u03b1+Bcos\u03b1=(A^2+B^2)^(1/2)cos(\u03b1-t)\uff0ctant=A/B
\u3000\u3000\u00b7\u500d\u89d2\u516c\u5f0f\uff1a
\u3000\u3000sin(2\u03b1)=2sin\u03b1\u00b7cos\u03b1=2/(tan\u03b1+cot\u03b1)
\u3000\u3000cos(2\u03b1)=cos^2(\u03b1)-sin^2(\u03b1)=2cos^2(\u03b1)-1=1-2sin^2(\u03b1)
\u3000\u3000tan(2\u03b1)=2tan\u03b1/[1-tan^2(\u03b1)]
\u3000\u3000\u00b7\u4e09\u500d\u89d2\u516c\u5f0f\uff1a
\u3000\u3000sin(3\u03b1)=3sin\u03b1-4sin^3(\u03b1)
\u3000\u3000cos(3\u03b1)=4cos^3(\u03b1)-3cos\u03b1
\u3000\u3000\u00b7\u534a\u89d2\u516c\u5f0f\uff1a
\u3000\u3000sin(\u03b1/2)=\u00b1\u221a((1-cos\u03b1)/2)
\u3000\u3000cos(\u03b1/2)=\u00b1\u221a((1+cos\u03b1)/2)
\u3000\u3000tan(\u03b1/2)=\u00b1\u221a((1-cos\u03b1)/(1+cos\u03b1))=sin\u03b1/(1+cos\u03b1)=(1-cos\u03b1)/sin\u03b1
\u3000\u3000\u00b7\u964d\u5e42\u516c\u5f0f
\u3000\u3000sin^2(\u03b1)=(1-cos(2\u03b1))/2=versin(2\u03b1)/2
\u3000\u3000cos^2(\u03b1)=(1+cos(2\u03b1))/2=covers(2\u03b1)/2
\u3000\u3000tan^2(\u03b1)=(1-cos(2\u03b1))/(1+cos(2\u03b1))
\u3000\u3000\u00b7\u4e07\u80fd\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1=2tan(\u03b1/2)/[1+tan^2(\u03b1/2)]
\u3000\u3000cos\u03b1=[1-tan^2(\u03b1/2)]/[1+tan^2(\u03b1/2)]
\u3000\u3000tan\u03b1=2tan(\u03b1/2)/[1-tan^2(\u03b1/2)]
\u3000\u3000\u00b7\u79ef\u5316\u548c\u5dee\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1\u00b7cos\u03b2=(1/2)[sin(\u03b1+\u03b2)+sin(\u03b1-\u03b2)]
\u3000\u3000cos\u03b1\u00b7sin\u03b2=(1/2)[sin(\u03b1+\u03b2)-sin(\u03b1-\u03b2)]
\u3000\u3000cos\u03b1\u00b7cos\u03b2=(1/2)[cos(\u03b1+\u03b2)+cos(\u03b1-\u03b2)]
\u3000\u3000sin\u03b1\u00b7sin\u03b2=-(1/2)[cos(\u03b1+\u03b2)-cos(\u03b1-\u03b2)]
\u3000\u3000\u00b7\u548c\u5dee\u5316\u79ef\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1+sin\u03b2=2sin[(\u03b1+\u03b2)/2]cos[(\u03b1-\u03b2)/2]
\u3000\u3000sin\u03b1-sin\u03b2=2cos[(\u03b1+\u03b2)/2]sin[(\u03b1-\u03b2)/2]
\u3000\u3000cos\u03b1+cos\u03b2=2cos[(\u03b1+\u03b2)/2]cos[(\u03b1-\u03b2)/2]
\u3000\u3000cos\u03b1-cos\u03b2=-2sin[(\u03b1+\u03b2)/2]sin[(\u03b1-\u03b2)/2]
\u3000\u3000\u00b7\u63a8\u5bfc\u516c\u5f0f
\u3000\u3000tan\u03b1+cot\u03b1=2/sin2\u03b1
\u3000\u3000tan\u03b1-cot\u03b1=-2cot2\u03b1
\u3000\u30001+cos2\u03b1=2cos^2\u03b1
\u3000\u30001-cos2\u03b1=2sin^2\u03b1
\u3000\u30001+sin\u03b1=(sin\u03b1/2+cos\u03b1/2)^2

\u89e3\u7b54\uff1a
\u4e00\u6b21\u51fd\u6570y=kx+b\u56fe\u8c61\u8fc7P\uff081\uff0c2\uff09\u3001
2=k+b....\u2460

\u4e00\u6b21\u51fd\u6570y=kx+b\u4e0e\u4e0ex\u8f74\uff0cy\u8f74\u7684\u4ea4\u70b9A,B\u5750\u6807\u5206\u522b\u4e3a\uff08-k/b\uff0c0\uff09\uff0c\uff080\uff0cb\uff09

tan\u2220PAO\uff1d1/2
\u2234\uff080-2\uff09/\uff08-k/b-1\uff09=1/2
\u89e3\u7b54\uff1ak=3b....\u2461

\u5c06\u2461\u4ee3\u5165\u2460\u4e2d\uff0c\u5f97\uff1ab=1/2

\u2234\u70b9B\u5750\u6807\u4e3a\uff081/2\uff0c0\uff09

（2）=（5+2tanA）/(3-tanA) （分子分母同时除cosA）
= （2+2*4）/（3-4）
=-10
（3）由（3）式的结论得A+B=90时 tanA*tanB=1
故=(tan1*tan89)*(tan2*tan88)*(tan3*tan87)*(tan4*tan86)=1*1*1*1=1

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