正四棱锥S-ABCD中,角ASB=45°,二面角A-SB-C为θ,且cosθ=m+√n,(m,n为整数),求m+n 如图，在正四棱锥P-ABCD中，∠APC=60°，则二面角A...
\u5df2\u77e5\u4e09\u68f1\u9525S-ABC\u4e2d,\u89d2ASB=\u89d2BSC=\u89d2CSA=\u03c0/2,\u6c42\u8bc1\u4e09\u89d2\u5f62ABS\u662f\u9510\u89d2\u4e09\u89d2\u5f62\u3002\u4f9d\u9898\u610f\u53ef\u5f97
AB^2=SA^2+SB^2\uff0c
AC^2=SA^2+SC^2\uff0c
BC^2=SB^2+SC^2\uff0c
2AB*BC*cos\u2220ABC=AB^2+BC^2-AC^2=2SB^2>0\uff0c
\u6240\u4ee5cos\u2220ABC>0\uff0c
\u540c\u7406\u53ef\u5f97cos\u2220ACB>0\uff0ccos\u2220BAC>0\uff0c
\u6240\u4ee5\u25b3ABC\u662f\u9510\u89d2\u4e09\u89d2\u5f62\u3002
\u8fc7A\u4f5cAE\u22a5PB\u4e8eE\uff0c\u8fde\u63a5EC\uff0cPO\uff0c\u8fde\u63a5AC\u3001BD\u4ea4\u4e8e\u70b9O \u2235PO\u662f\u6b63\u56db\u68f1\u9525P-ABCD\u7684\u9ad8\uff0cPO\u22a5\u9762ABCD\uff0cAC?\u5e73\u9762ABCD\u2234AC\u22a5PO\u53c8\u2235\u6b63\u65b9\u5f62ABCD\u4e2d\uff0cAC\u22a5BD\uff0cPO\u3001BD\u662f\u5e73\u9762PBD\u5185\u7684\u76f8\u4ea4\u76f4\u7ebf\u2234AC\u22a5\u5e73\u9762PBD\uff0c\u5f97PB\u22a5AC\u2235AE\u22a5PB\uff0cAC\u3001AE\u662f\u5e73\u9762ACE\u5185\u7684\u76f8\u4ea4\u76f4\u7ebf\u2234PB\u22a5\u5e73\u9762ACE\uff0c\u5f97CE\u22a5PB\u56e0\u6b64\uff0c\u2220AEC\u662f\u4e8c\u9762\u89d2A-PB-C\u7684\u5e73\u9762\u89d2\u8bbeAB=1\uff0c\u5f97AC= 2 \u2235\u6b63\u56db\u68f1\u9525P-ABCD\u4e2d\uff0cPA=PC\uff0c\u2220APC=60\u00b0\uff0c\u2234\u25b3ACP\u662f\u6b63\u4e09\u89d2\u5f62\uff0c\u5f97PA=PC=AC= 2 \u25b3PAB\u4e2d\uff0ccos\u2220PBA= A B 2 +P B 2 -P A 2 2\u00d7AB\u00d7PB 2 4 \u2234Rt\u25b3ABE\u4e2d\uff0cBE=ABcos\u2220PBA= 2 4 \uff0cAE= A B 2 -B E 2 = 14 4 \uff0c\u540c\u7406\u5f97\u5230CE= 14 4 \uff0c\u25b3AEC\u4e2d\uff0ccos\u2220AEC A E 2 +C E 2 -A C 2 2AE\u00d7CE =- 1 7 \uff0c\u5373\u4e8c\u9762\u89d2A-PB-C\u7684\u5e73\u9762\u89d2\u7684\u4f59\u5f26\u503c\u4e3a- 1 7 \uff0c\u6545\u9009\uff1aB
角ASB=45°=Pi/4, 角SAB=SBA=3/8Pi, 设AB=aSB/sin(3/8Pi) = AB /sin(Pi/4) , SB=2^2 sin(3/8Pi) a
设AP垂直SB于P, 易知CP垂直SB, AP=CP=SP=asin(Pi/4)=a/2^2
cosθ=(AP^2 + CP^2 - AC^2) /( 2AP*CP)=-3+2*2^2
m=-3, n=8
m+n=5
绛旓細瑙扐SB=45掳=Pi/4, 瑙SAB=SBA=3/8Pi, 璁続B=a SB/sin(3/8Pi) = AB /sin(Pi/4) , SB=2^2 sin(3/8Pi) a 璁続P鍨傜洿SB浜嶱, 鏄撶煡CP鍨傜洿SB, AP=CP=SP=asin(Pi/4)=a/2^2 cos胃=(AP^2 + CP^2 - AC^2) /( 2AP*CP)=-3+2*2^2 m=-3, n=8 m+n=5 ...
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绛旓細E鍦ABCD鐨鍨傜洿鎶曞奖涓篊O鐨勪腑鐐逛负E1锛岃SC=1锛屽垯CO=鏍瑰彿锛2锛/2锛孲鍒癇C涓偣F鐨勮窛绂籗F=鏍瑰彿锛3锛/2锛孲OF涓虹洿瑙掍笁瑙掑舰锛孲O= 鏍瑰彿锛2锛/2 EE1=鏍瑰彿锛2锛/4锛BE1=鏍瑰彿锛圔O^2+OE1^2) = 鏍瑰彿锛3/8锛塼an<EBE1 = 1/鏍瑰彿锛3锛<EBE1 = 30搴 sin<EBE1 = 1/2涓烘墍姹 ...
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