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heureka

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 #1
avatar+26396 
+1

(tan15∘)x(tan25∘)x(tan35∘)x(tan85∘)

 

Formula:

sin(x)sin(y)=12(cos(xy)cos(x+y))cos(x)cos(y)=12(cos(xy)+cos(x+y))sin(2x)=2sin(x)cos(x)

 

tan(15)tan(25)tan(35)tan(85)=sin(15)sin(25)sin(35)sin(85)cos(15)cos(25)cos(35)cos(85)=[sin(85)sin(15)][sin(35)sin(25)][cos(85)cos(15)][cos(35)cos(25)]sin(85)sin(15)=12(cos(8515)cos(85+15))sin(85)sin(15)=12(cos(70)cos(100))sin(35)sin(25)=12(cos(3525)cos(35+25))sin(35)sin(25)=12(cos(10)cos(60))cos(85)cos(15)=12(cos(8515)+cos(85+15))cos(85)cos(15)=12(cos(70)+cos(100))cos(35)cos(25)=12(cos(3525)+cos(35+25))cos(35)cos(25)=12(cos(10)+cos(60))=12(cos(70)cos(100))12(cos(10)cos(60))12(cos(70)+cos(100))12(cos(10)+cos(60))=(cos(70)cos(100))(cos(10)cos(60))(cos(70)+cos(100))(cos(10)+cos(60))|cos(100)=cos(90+10)=sin(10)=(cos(70)+sin(10))(cos(10)cos(60))(cos(70)sin(10))(cos(10)+cos(60))|cos(70)=cos(9020)=sin(20)=(sin(20)+sin(10))(cos(10)cos(60))(sin(20)sin(10))(cos(10)+cos(60))|cos(60)=12=(sin(20)+sin(10))(cos(10)12)(sin(20)sin(10))(cos(10)+12)=sin(20)cos(10)12sin(20)+sin(10)cos(10)12sin(10)sin(20)cos(10)+12sin(20)sin(10)cos(10)12sin(10)|sin(10)cos(10)=12sin(20)=sin(20)cos(10)12sin(20)+12sin(20)12sin(10)sin(20)cos(10)+12sin(20)12sin(20)12sin(10)|12sin(20)12sin(20)=0=sin(20)cos(10)+012sin(10)sin(20)cos(10)+012sin(10)=sin(20)cos(10)12sin(10)sin(20)cos(10)12sin(10)=1

 

laugh

29.03.2017
 #3
avatar+26396 
+2

knowing that Tan(45)=1, find Sin(45) and Cos(45) with the fonamental relations

 

tan(45)=1cot(45)=1tan(45)=11=1

 

Formula:

sin2(x)+cos2(x)=1

 

sin2(45)+cos2(45)=1|:cos2(45)sin2(45)+cos2(45)cos2(45)=1cos2(45)sin2(45)cos2(45)+cos2(45)cos2(45)=1cos2(45)tan2(45)+1=1cos2(45)|tan(45)=11+1=1cos2(45)2=1cos2(45)|square root both sides2=1cos(45)12=cos(45)

 

sin2(45)+cos2(45)=1|:sin2(45)sin2(45)+cos2(45)sin2(x)=1sin2(45)sin2(45)sin2(45)+cos2(45)sin2(45)=1sin2(45)1+cot2(45)=1sin2(45)|cot(45)=11+1=1sin2(45)2=1sin2(45)|square root both sides2=1sin(45)12=sin(45)

 

laugh

29.03.2017
 #4
avatar+26396 
+3

Let ABCDEFGH be a cube of side length 5, as shown.

Let P and Q be points on line AB and line AE, respectively, such that AP = 2 and AQ = 1.

The plane through C, P, and Q intersects line DH at R. Find DR. 

 

 

Let C=0,0,0Let P=5,3,0Let Q=5,5,1Let DR=zLet R=0,5,z

 

The plane through C, P, and Q:

x=C+s(PC)+t(QC)|x=RC=0R=0+s(P0)+t(Q0)R=sP+tQ|P=(530)Q=(551)R=(05z)(05z)=s(530)+t(551)

(1)0=5s+5t0=s+ts=t(2)5=3s+5t5=3(t)+5t5=2tt=52(3)z=0s+tz=tz=52

 

DR = 52

 

laugh

28.03.2017