Kernestof
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b (B) (B)
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C )
a(A ) b(B) =⋅ a =b = + c − ⋅b⋅c a(A ) b(B) =⋅ a =b = + c − ⋅b⋅c a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
c(C ) (A)
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a (A ) (A )
x = x = xx == x =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( )) ( )
a = a = aa ==
=
−
= va =
=
Serien Kernestof findes i forskellige udgaver rettet mod stx, hf, hhx og htx:
c(C ) (A) c (A) (B)
− b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a c b a vv == =−− = b a =a b + c − ⋅a b⋅c ⋅ (B) (A) (C ) (B) (A ) c = a + b − ⋅a⋅b⋅ (C ) b c a =− = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under(B) (C ) c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) H:v ind Cosinusrelationerne b = ++ cb −
c b − ⋅a⋅b (C ) b på = abagflappen. +c c⋅ (B) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A ) (B) (C ) c =(A) a= + b − ⋅a⋅b⋅ (C ) = = Kvadratsætningerne ⋅b⋅c a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne (a + b)
⋅b⋅c c =(A) a =+ b b + −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − (a − b) a + b − ab
a = b + c= ⋅c + ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
b +c − a ⋅b⋅c b +c − a (A) = ⋅b⋅c (A) =
=
+ ⋅ −
=
+
−
=
−
=
+ ⋅ −
=
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
− + = − ( − ) + = + −
+
= − = +
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b (B) (B)
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C )
a(A ) b(B) a =b = + c − ⋅b⋅= c⋅ a(A ) b(B) a =b = + c − ⋅b⋅= c⋅ a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
c(C ) (A)
b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ a
(B) (A)
b a =a b + c − ⋅a b⋅c ⋅ c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A a ) = = a (A ) (A )
(B) (A) (C ) (A)
x = x = xx == x =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( b(B)))=
a = a = aa ==
=
−
= va =
−
v = a −− vv == = (A ) a
b
=−
c (C ) c
=
=
c(C ) (A)
(a + b)
⋅b⋅c c =(A) a =+ b b + −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − a + b − ab (a − b)
a = b + c= ⋅c+ ⋅ ab (A) (a− + b) = −a ⋅+bb − ab (a b) == aa ++bb −− ab ab (a (a−−b) b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) b) +⋅bc ⋅− ab (a +−b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
(A) = (A) =
(B )=
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b (B) (B)
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C ) c(C ) (A)
x =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( b(B)))=
a = a =
=
−
= va =
− v = a −− vv == = (A ) a
=−
b
=
=
+ ⋅√ + ⋅√
c(C ) (A) c (A) (B)
− + ⋅+√ − + ⋅+√
(B) (A)
b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ a
c (C ) c
(B) (A) (C ) (A)
b a + c − ⋅a a =b b⋅c ⋅ c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
−
(A )
=
(a + b)
(B)
=
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
c =(A) a =+ b b + −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − a + b − ab (a − b)
a = b + c= ⋅c + ⋅ ab (A) (a− + b) = −a ⋅+bb − ab (a b) == aa ++bb −− ab ab (a (a−−b) b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
−
(A) =
(B )= (B )=
(C ) = (C ) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+ ⋅ −
=
+ +⋅ −= = + +⋅ −= =
= − = +
+
= − = +
( − ) +
+ =
= +
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
(
)⋅ +
−
=
+√ = − − )⋅ +
−
(
= + √ −= −= − −
−
+ ⋅√
=
(
−
)⋅ +
(
−
)⋅ +
+ ⋅√
⋅ +
+
=
⋅ +
+
=
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
− + ⋅+√ − + ⋅+√ −
=
−
−
± ′
− x +==− =− − =− −
x =−
x ==−−
−
=
−
=
= −
− =
= −
− =
(
−
+√ = − − )⋅ +
=
= + √ −= −= − −
− += − = − − − + =− − − − = =− − −= − − − + −= − − − − += −− = − − −
x = −= x =−
x− −= − = −−
⋅(x − )=
−
=
−
x⋅= (x − )=
− −
∫
∫
∫
x⋅= (x − )= x⋅= (x − )=
x = − eller x =
(x + )⋅( − x ) =
x = − eller x =
a(A ) b(B) =c − ⋅b⋅= a =b + c⋅ a(A ) b(B) =c − ⋅b⋅= a =b + c⋅ a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
c(C ) (A)
x = xx == x =
a +c − b ⋅a⋅c a +c − b (B )= ⋅a⋅c
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( )) ( )
a = a = aa ==
=
−
= va =
=
c(C ) (A) c (A) (B)
− b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a c b a −− vv == = = b a + c − ⋅b a⋅c ⋅ (B) a =b (A) (C ) (B) (A ) c = a + b − ⋅a⋅b⋅ (C ) a b c =− = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under(B) (C ) c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) H:v ind Cosinusrelationerne b = ++ cb −
c = abagflappen. +b b⋅ (C b på c − ⋅a⋅c (B)) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A )
(B)
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind på c = abagflappen. + b − ⋅a⋅b⋅ (C ) (A) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c =
(a + b)
=
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a (a − b) c= −a ⋅+bb − ab
a = b + c= ⋅c+ ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab b) == aa ++bb −− ab ab (a (a−−b) b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
(A) = (A) =
(B )= (B )=
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a
x = x =
x =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b
x = − eller x =
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c
x = − eller x =
(C ) (B) (A ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) =c − ⋅b⋅c =⋅ a =b + (A) a(A ) b(B) c(C ) (( ( ))) =c − ⋅b⋅c =⋅ a =b + (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) = va = b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b a + c − ⋅b a⋅c ⋅ (B) ))= (C ) a =b (A) (A ) (B) c = a + b − ⋅a⋅b⋅ (C ) a b c = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb − a =
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
−
−
−−
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
−
c = abagflappen. +b (C b på c − ⋅a⋅b c⋅ (B)) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A )
=
(a + b)
(B)
=
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
= a + b + ab
(a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − a + b − ab (a − b)
a = b + c= ⋅c + ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
− + ⋅+√ −
+
= − = +
+ =
= +
=
−
+
= − = +
+ =
= +
=
( − ) + ( − ) +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √ (
(B )=
(C ) =
)⋅ +
−
+√ = − − )⋅ +
−
(
=
= + √ −= −= − −
−
(
−
)⋅ +
(
−
)⋅ +
− += − = − − − + =− − − − = =− − −= − − − + −= −
=
−
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
⋅ +
= =
+ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
− x +==− =− − =− −
x ==−−
+ =
= +
+ =
= +
=
=
( − ) +
=
=
=
=
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
−
(
+
(
= − = +
)⋅ +
−
− −
+√ = − − )⋅ +
=
= + √ −= −= − −
− += − = − − − + =− − − − = =− − + −= − −= − − − − − − += −− = − − −
x = −=
=−
x− −= − = −− −
x ==−− ⋅(x − )=
= − = +
+
( − ) +
=− −− + =−= + =− −
−
x =−
−
=
+= − − =− − + −=x −= x − − x
−
x−= −= −
− + ⋅+√
=
+ −=x −= x − − x + = x−
x− −= − = −−
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
⋅ +
+ −= −= − − + −=x x=− x − − x
− − − += −− = − − −
x = −=
−
(B )=
(C ) =
=
−
x =−
− −
=
(
−
)⋅ +
(
−
)⋅ +
−
=
+
−
=
+
−
=
+
−
=
=
)⋅ +
x= x = − eller x = (x + )⋅( − x ) =
x(x = + − )⋅( eller − x ) x= = x = − eller x = x = − eller x =
=− + =−− = −= − − +
=−
x =−
⋅ + ⋅ +
= =
+ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x + −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x += − − =− − + −=x −= x − − x =− −− + =−= + =− −
=−
x =−
=
′
=
=
=
=
=
=
=
=
(
−
−
+√ = − − )⋅ +
(
)⋅ +
=
= + √ −= −= − −
−
(
−
)⋅ +
(
−
)⋅ +
=
+ +
=
=
=
=
=
=
=
=
⋅ +
+
=
⋅ +
+
=
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
±
− += − = − − − + =− − − − = =− − −= − − − + −= −
=
−
′
′
′
± ′
+ −= −= − − + −=x x=− x − − x += − − =− − + −=x −= x − − x =− + =−− = −= − − +
−
−
′
+ −=x −= x − − x + = x−
−
x− = −= −
− x +==− =− − =− −
x =−
x ==−− =−
′
x ==−−
⋅(x − )=
a = a =
aa ==
x =−
−
= va =
−
−−
+ ⋅√
−
+ ⋅√
c = abagflappen. +b b⋅ (B) (C ) b på c − ⋅a⋅c H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A )
(B)
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
b + c =(A) a =+ b −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
− + ⋅+√
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c =
(a + b)
=
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − (a − b) a + b − ab
a = b + c= ⋅c + ⋅ ab (A) (a− + b) = −a ⋅+bb − ab (a (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) b) +⋅bc ⋅− ab (a +−b)(a − b) = a − b (a (a++b)(a b)(a−−b) b) == aa −−bb c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
− + ⋅+√ − −
−
=
−
=
= −
− =
= −
− =
+
( − ) +
x =
x =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( b(B)))=
a = a =
−
=−
b
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a (A )
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b (B)
(A )
(B)
=
=b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ ab =
c (C ) c
=
(A )
(B)
=
b a + +c − a = =b − ⋅ ⋅a b⋅ ⋅c ⋅ ⋅ c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
(C )
=
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
= a + b + ab
b + c =(A) a =+ b −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a ⋅a⋅c (A) = a +c − cb (B )= b +⋅cb⋅− a (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b ⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c cb (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = bc a +⋅ba⋅− (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(a (a++=b) b)b +== aa ++bb c++⋅ ab ab(A) a b) c= −a ⋅+b⋅ b − ab (a −
a = b + c= ⋅c − ⋅ ab (A) (a− + b) + (a = −a ⋅+bb ab (a (a−−b) b) == aa ++bb −− ab ab b a +c − ⋅= a⋅ca⋅ − b(B) (a +=b)(a − b) b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
(B )= (B )=
(C ) = (C ) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
c(C ) (A) c(C ) (A) c (A) (B) (B) (A) (B) (A) (C ) (A)
+ ⋅√
−
+ ⋅√
=
−
− + ⋅+√
(C )
+ ⋅ −
=
= −
− + ⋅+√
=
+ ⋅ −
− =
= −
+
=
−
+
+ +⋅ −= =
− =
=
−
+
+ +⋅ −= =
=
−
+
−
+
( − ) + −
+
( − ) +
= − = +
+ =
= − = +
= +
+ =
(
)⋅ +
−
(
=
+√ = − − )⋅ +
(
= + √ −= −= − −
⋅ +
+
=
⋅ +
=
+
=
=
=
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
)⋅ +
−
(
=
=
′
=
′
′ ′
± ′
+ −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x
′
+= − − =− − + −=x −= x − − x
±
′
′ ′
−
± ′
= =− − + + −= −= − −
− −
− += − = − − − + =− − − − = =− − −= − − − + −= −
x = − eller x =
∫
∫
∫
=
±∫
− − − += −− = − − −
x = −= x =
x =
( ) (( )) (( ( ))) ((( )) ) ( ) ( ) ( ) (( b(B)))=
a = a = aa ==
b
=−
=
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b (B) (B)
=
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c (C ) c c (C ) c(C ) (A)
−
c(C ) (A) c (A) (B) (B) (A)
b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ a
c (C ) c
x− −= − = −−
(C )
=
a(A ) b(B) =c − ⋅b⋅c =⋅ a =b + a(A ) b(B) =c − ⋅b⋅c =⋅ a =b + a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c⋅
−
= va =
− v = a vv == =−− (A ) a
(B) (A) (C ) (A)
b a + c − ⋅a a =b b⋅c ⋅ c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
+ ⋅√
= (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind b = ++ cb −
+ ⋅√
c +b (C ) b på = abagflappen. c − ⋅a⋅b c⋅ (B) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne H: H: ind ind under Cosinusrelationerne b på på = abagflappen. bagflappen. + c − ⋅a ⋅c ⋅ (B) a under b Cosinusrelationerne c b +c − a (A )
(B)
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
− + ⋅+√
(C )
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind på c = abagflappen. + b − ⋅a⋅b⋅ (C ) (A) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c =
(a + b)
=
= a + b + ab
b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅bb⋅− (C ) = a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = bc a +⋅ba⋅− (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a (a − b) c= −a ⋅+bb − ab
a = b + c= ⋅c+ ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +c − ⋅= a⋅ca⋅ − b(B) − b) (a +=b)(a
Kvadratsætningerne
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
x =
(A) = (A) =
x = a = a =
va = =
( ) (( )) (( ( ))) (( )) ) ( ) ( ) ( ) (( ))
(B )= (B )=
−
−−
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
− + ⋅+√ −
(C ) =
−
=
−
=
= −
− =
= −
− =
+
( − ) + −
+
( − ) +
+ ⋅ −
=
+ ⋅ −
=
+ +⋅ −= = + +⋅ −= =
= − = +
+ =
= − = +
= +
+ =
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
Kvadratsætningerne
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne Kvadratsætningerne
(
)⋅ +
−
−
(
−
+√ = − − )⋅ +
(
=
= + √ −= −= − −
−
+
−
+
−
+
−
=
−
= =
= =
− −
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
Kvadratsætningerne
⋅ +
+
=
⋅ +
=
+
=
=
=
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
)⋅ +
−
=
=
=
+ ⋅ −
=
+ +⋅ −= = + +⋅ −= =
+ =
= +
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
⋅(x − )=
= =
−
)⋅ +
−
)⋅ +
=
⋅ + ⋅ +
= =
+ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
x⋅= (x − )=
+ −=x −= x − − x + = x−
Lindhardt og Ringhof
+= − − =− − + −=x −= x − − x =− + =−− = −= − − +
x ==−−
x⋅= (x − )=
=−
=
+ −= −= − − + −=x x=− x − − x
x ==−−
+= − − =− − + −=x −= x − − x
x− = − − − =− −
+ −= −= − − + −=x x=− x − − x
− x +==− =− − =− −
x =−
=
=
+ −=x −= x − − x + = x−
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne
−
(
x− = −= −
=
)⋅ +
−
(
− − − += − − −− = − −
x = −=
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
= =− − + + −= −= − −
− −
− x +==− =− − =− −
x− = −= −
x ==−−
x =−
=−
= under H:v ind på bagflappen. ( Cosinusrelationerne )
x ==−−
−
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
⋅(x − )=
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab
⋅(x − )=
x =−
⋅(x − )=
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H:ind indunder underCosinusrelationerne Cosinusrelationernepå påbagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne Kvadratsætningerne
+ ⋅√
Kvadratsætningerne (a + b) = a + b + ab
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+
−
=
+
−
=
+
−
=
+
−
=
x =−
x⋅= (x − )= x⋅= (x − )=
(a (a+− +b) b) ab b) === aaa+++bbb++− ab ab (a− + b) (a
+
=
= +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + = ( − )⋅ + = = √
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind indunder underCosinusrelationerne Cosinusrelationernepå påbagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne
− += − − =− − − + =− − − − = =− − + −= − −= − − −
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
Kvadratsætningerne
(C ) =
−
v =
(C )
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind indunder underCosinusrelationerne Cosinusrelationernepå påbagflappen. bagflappen.
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne
b(a= a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b (a (a b)(a−−b) == aa − −bb c ++=b)(a a + b b) − ⋅a⋅b⋅ (C )
x =
xx ==
aa ==
vv ==
(
′
x =−
⋅(x − )=
x = − eller x =
∫ ±
)⋅ +
=−
x ==−− ⋅(x − )=
x(x = + − )⋅( eller − x ) x= =
±∫
= + √ −= −= − −
x ==−−
x =−
∫
+√ = − − )⋅ +
′
− x +==− =− − =− −
x− = −= −
x⋅= (x − )= x= (x + )⋅( − x ) =
∫
∫
−
−
=
=
±
=
−
−
(
′
=
)⋅ +
−
(
− − − += −− = − − −
x− −= − = −−
x⋅= (x − )=
(x + )⋅( − x ) =
∫ ±
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a a(A ) = = a (A ) (A )
x =
xx ==
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
+ ⋅ −
=
=
= +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + = ( − )⋅ + = = √
− −
− += − = − − − + =− − − − = =− − + −= − −= − − −
x = −=
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
= − = +
′
=
−
(C )
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
= − = +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c (C ) c c (C ) (C )
=
a(A ) b(B) =c − ⋅b⋅= a =b + c⋅ a(A ) b(B) =c − ⋅b⋅= a =b + c⋅ a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
−
= va =
(a + b)
Kvadratsætningerne (a + b) = a + b + ab
x= x = − eller x = (x + )⋅( − x ) =
(B )=
(C ) =
+
( − ) +
x =
xx ==
aa ==
v = a −− vv == = (A ) a
= (A ) under =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind bc på = ++ cb −
c +b b⋅ (B) (C ) b på = abagflappen. c − ⋅a⋅c H: ind Cosinusrelationerne (A ) under(B) (C ) Kvadratsætningerne = = H: H: ind ind under Cosinusrelationerne b på på = abagflappen. bagflappen. +b c + −c ⋅− a⋅ac ⋅ (B) a under b Cosinusrelationerne c
Kvadratsætningerne a under b Cosinusrelationerne c b +c −a H: ind på c = abagflappen. + b − ⋅a⋅b⋅ (C ) (A) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(B )=
(C ) =
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
b) == aa ++bb −− ab ab (a (a−−b) (a + b)(a − b) = a − b
− + + ⋅√
(a − b) = a + b − ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a
+ ⋅ −
=
+ = + ⋅ − = = +
+ =
+ =
= +
(x + )⋅( − x ) = x= (x + )⋅( − x ) =
+ = + ⋅ − =
( −− + ) + == − +
x= x(x = − eller x = + )⋅( − x ) =
fp_k=VTUJUTJTMSSJUSTJR
( −− + ) + == − +
⋅(x − )=
x(x = + − )⋅( eller − x ) x= =
=
(a + b)(a − b) = a − b
(x + )⋅( − x ) =
x = − eller x =
=
x = − eller x = ( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
+
=
)⋅ +
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
− )⋅ + + =−
√
=
=− + − =− − + = −− = − = −− − − − + + −= −= − −
− −
=− = −
− −
− −
+ == x⋅− +
+
=
=
=
=
=
=
=
=
xx = =
a = aa == va = =
x =−
=− − =− − += −
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( ))
−
v =
−
vv ==
−−
x⋅= (x − )=
= under H:v ind på bagflappen. ( Cosinusrelationerne ) −
x =−
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
=− − =
x =−
x⋅= (x − )=
x = a =
+ =− + − x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
= − = −− − − + − =− −
x− = − − x = −=
=
x =
−
(
−
x =
+
=
(
( −+ − )⋅ += − − ( − = − +√ = − − − − −
x− = − −
x ==−−
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab
⋅(x − )=
x =−
+ ⋅√
Kvadratsætningerne (a + b) = a + b + ab
⋅(x − )=
(a (a+− +b) b) ab (a b) === aaa+++bbb++− ab ab
x= ⋅(x − )=
(a− + b) (a
x= (x⋅(x + − )⋅( )= −x ) =
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
(a (a−−b) b) == aa ++bb −− ab ab (a + b)(a − b) = a − b
− + + ⋅√
(a − b) = a + b − ab (a + b)(a − b) = a − b
x= (x + )⋅( − x ) =
(a (a++b)(a b)(a−−b) b) == aa −−bb
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+ ⋅ −
=
+ = + ⋅ − = + = + ⋅ − =
( −− )+ + == − +
x == − eller x =
= +
+ =
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
(x + )⋅( − x ) = x= (x + )⋅( − x ) =
=
(a + b)(a − b) = a − b
(x + )⋅( − x ) =
( −− )+ + == − +
x = − eller x = (x + )⋅( − x ) =
=
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
x = − eller x = x = − eller x =
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
+
=
+
=
(
−
)⋅ +
( −+ − )⋅ += − − ( − = − +√ = −
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
(
−
− )⋅ + + =−
√
=
⋅ + + == x −
− − − −
=− + − =− − + = −− = − = −− − − − + + −= −= − − = − = −− − − + − =− −
x− = − − − −
=− = −
− −
− −
x = −=
+
=
=
=
=
=
=
=
=
=
x= (x + )⋅( − x ) =
x= x = − eller x = (x + )⋅( − x ) =
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x− = − −
x =−
=− − =− − += −
x =−
=− − =
x =−
x(x = + − )⋅( eller − x ) x= =
x ==−− ⋅(x − )=
x =− ⋅(x − )=
x= ⋅(x − )= x= (x⋅(x + − )⋅( )= −x ) =
Per Gregersen og Majken Sabina Skov
x= (x + )⋅( − x ) =
x == − eller x =
x = − eller x =
(x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) = x = − eller x = x = − eller x =
x = − eller x =
Lindhardt og Ringhof
x= x = − eller x = (x + )⋅( − x ) =
∫ ±
∫
∫
∫
x(x = Per Gregersen Henrik + − )⋅(&eller − x ) x= =Bindesbøll Nørregaard
±∫
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a a(A ) = = a (A ) (A )
x = x = xx == x =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( b(B)))=
a = a = aa ==
−
=−
b
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b (B) (B)
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C )
a(A ) b(B) =c − ⋅b⋅c =⋅ a =b + a(A ) b(B) =c − ⋅b⋅c =⋅ a =b + a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
c(C ) (A)
=
−
= va =
v = a vv == =−− (A ) a
=
=b a ++cc − − ⋅⋅b a⋅⋅cc⋅⋅ ab =
c (C ) c
=
b a + +c − − ⋅ ⋅a ⋅c ⋅ a = =b b⋅ ⋅ c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
(A )
(B)
=
(a + b)
(C )
=
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
= a + b + ab
+ c ⋅− c =(A) a =+ b b − a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c cb (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = bc a +⋅ba⋅− (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − (a − b) a + b − ab
a = b +c ⋅c + ⋅ ab (A) (a− + b) = = −a ⋅+bb − ab (a (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) b) +⋅bc ⋅− ab (a +−b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
(B )= (B )=
(C ) = (C ) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
Praxis
c(C ) (A) c (A) (B) (B) (A) (B) (A) (C ) (A)
+ ⋅√
= (A ) under =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind bc på = ++ cb −
−
+ ⋅√
c +c b − ⋅a⋅b (C ) b på = abagflappen. c⋅ (B) H: ind Cosinusrelationerne (A ) under(B) (C ) Kvadratsætningerne = = b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a
=
−
− + ⋅+√
(C )
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind på c = a bagflappen. + b − ⋅a⋅b⋅ (C ) (A) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c
+ ⋅ −
=
= −
− + ⋅+√
=
+ ⋅ −
− =
= −
+
=
−
+
+ +⋅ −= =
− =
=
−
+
+ +⋅ −= =
=
−
+
′
=
−
x = − eller x =
=
(C ) −
+
( − ) + −
+
( − ) +
= − = +
+ =
= − = +
= +
+ =
=
= +
′
=
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + = ( − )⋅ + = = √ (
)⋅ +
−
(
(
=
+√ = − − )⋅ +
−
= + √ −= −= − −
−
)⋅ +
−
(
)⋅ +
−
⋅ +
+
=
⋅ +
=
+
=
=
=
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
′
=
± − += − = − − − + =− − − − = =− − + −= − −= − − −
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
=
−
′ ′
± ′
+ −= −= − − + −=x x=− x − − x
′
+= − − =− − + −=x −= x − − x
x− −= − = −−
±
′
′ ′
± ′
= =− − + + −= −= − −
− −
−
′
+ −=x −= x − − x + = x−
− − − += −− = − − −
x = −=
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
′
− x +==− =− − =− −
x− = −= −
x ==−−
x =−
=−
x ==−− ⋅(x − )=
′
x =−
⋅(x − )=
x⋅= (x − )= x⋅= (x − )= (x + )⋅( − x ) = x= (x + )⋅( − x ) = x= x = − eller x = (x + )⋅( − x ) =
∫ ±
∫
∫ ∫
x(x = + − )⋅( eller − x ) x= =
±∫
x = − eller x =
∫ ±
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a (A ) (A )
x = x = xx == x =
( ) (( )) (( ( ))) (( )) ) ( ) ( ) ( ) (( b(B)))=
a = x =
a = aa ==
x = xx ==
a =
x = x = xx == x =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b (B) (B)
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C )
va = =
=
−
−−
b
=−
=
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b (B) (B) =
a(A ) b(B) a =b = + c − ⋅b⋅= c⋅ a(A ) b(B) a =b = + c − ⋅b⋅= c⋅ a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
−
v = a −− vv == = (A ) a
( ) (( )) (( ( ))) (( )) ) ( ) ( ) ( ) (( ))
c (C ) c
∫
x = − eller x =
∫ ∫
±∫
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c (C ) c c (C ) (C ) c(C ) (A) c(C ) (A) c (A) (B)
(A )
(B)
(C )
(B) (A) (C ) (A)
b a = =a b + +c − − ⋅ ⋅a b⋅ ⋅c ⋅ ⋅ c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
+ ⋅√ + ⋅√ − + ⋅+√
(C )
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c =
(a + b)
=
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab
−
(a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a b) c= −a ⋅+bb − ab (a −
a = b + c= ⋅c + ⋅ ab (A) (a (a− + b) = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +c − ⋅= a⋅ca⋅ − b(B) (a +=b)(a − b)
c =(A) a =+ b b + −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a c (A) = a +⋅ca⋅− cb (B )= b +⋅cb⋅− a (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b ⋅− (C ) = a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a +b − c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = bc a +⋅ba⋅− (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
= −
− =
+ = + ⋅ − =
+
−
=
= −
− =
+ = + ⋅ − =
+
−
=
b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
x =
(a +b) b) ab (a+− b) === aaa+++bbb++− ab ab = = a +b + − ab ab
(a (a−−b) b) == aa ++bb −− ab ab (a + b)(a − b) = a − b b) −=b)a = + b − ab (a+−b)(a a −b (a b)(a−−b) b) == aa −−bb (a (a++b)(a
− + + ⋅√
+ =
=
+ =
= +
=
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
⋅ +
=
=
+
+
−
)⋅ +
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
(
− )⋅ + + =−
√
=
=− = − − −
=
v =
(A) =
(B )= (B )=
−
⋅ + + == x −
=
+
=
=
=
=
=
=
x =−
=− − =− − += −
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
− + ⋅+√ −
−
=
−
=
= −
− =
= −
− =
+
( − ) + −
+
( − ) +
+ ⋅ −
=
+ ⋅ −
=
+ +⋅ −= = + +⋅ −= =
= − = + = − = +
+ =
= +
+ =
= +
Kvadratsætningerne
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne Kvadratsætningerne
(
−
−
+√ = − − )⋅ +
(
−
)⋅ +
=
= + √ −= −= − −
−
=
+
−
=
+
−
=
+
−
=
=
(
−
)⋅ +
(
−
)⋅ +
− += − = − − − + =− − − − = =− − −= − − − + −= −
Kvadratsætningerne
⋅ + ⋅ +
= =
+ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
+ −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x
− − − += −− = − − −
x = −=
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
+= − − =− − + −=x −= x − − x
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne Kvadratsætningerne
x− −= − = −−
= =− − + + −= −= − −
− − x−= −= −
− x +==− =− − =− −
−
=
−
Henrik Bindesbøll Nørregaard og Per Gregersen
x ==−−
x =−
=−
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
x ==−−
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab
⋅(x − )=
+ ⋅√
Kvadratsætningerne (a + b) = a + b + ab
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+
−
=
+
−
=
+
−
=
+
−
=
x =−
⋅(x − )=
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne Kvadratsætningerne
(a (a+− +b) b) ab b) === aaa+++bbb++− ab ab (a (a− + b) (a
+
=
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + = ( − )⋅ + = = √
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne
(C ) =
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
b) == aa ++bb −− ab ab (a (a−−b) (a + b)(a − b) = a − b
− + + ⋅√
(a − b) = a + b − ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a
x =−
+ ⋅ −
=
+ = + ⋅ − = = +
+ =
+ =
= +
x⋅= (x − )= x⋅= (x − )=
+ = + ⋅ − =
( −− + ) + == − +
=− − =
x =−
b +c −a ⋅b⋅c b +c −a ⋅b⋅c
a + b −c ⋅a⋅b a + b −c (C ) = ⋅a⋅b
−
−−
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
= − = −− − − + − =− −
x− = − − − −
va = =
=
−
(
( −+ − )⋅ += − − ( − = − +√ = − − =− − + − =− − − − + = −− = − = −− − − − + + −= −= − −
x− = − −
− −
( ) (( )) (( ( ))) (( )) ) ( ) ( ) ( ) vv == (( )) = under H:v ind på bagflappen. ( Cosinusrelationerne ) a =
aa ==
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
x = −=
(A) =
x =
= +
( −− + ) + == − +
−
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
x = xx ==
a =
( −− + ) + == − +
(a + b)(a − b) = a − b
(C )
H: ind under Cosinusrelationerne på bagflappen. H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne Kvadratsætningerne
+ ⋅√
− + + ⋅√
Kvadratsætningerne (a + b) = a + b + ab
x = − eller x =
(B) (A)
=b a ++ cc − − ⋅ ⋅b a⋅⋅cc⋅⋅ ab =
= (A ) under =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind bc på = ++cb −
c b − ⋅a⋅cb⋅ (B) (C ) b på = abagflappen. +c H: ind Cosinusrelationerne (A ) under(B) (C ) Kvadratsætningerne = = b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a
−
v = vv ==
= under H:v ind på bagflappen. ( Cosinusrelationerne )
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab
(a− + b) (a
−
= va =
x = a =
aa ==
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A a ) = = a (A ) (A )
(x + )⋅( − x ) = x= (x + )⋅( − x ) = x= x = − eller x = (x + )⋅( − x ) =
=
(a + b)(a − b) = a − b x ==−−
( −− + ) + == − +
x =−
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = √
x(x = + − )⋅( eller − x ) x= =
=
⋅(x − )=
fp_k=VTUJUTJTMSSJUSTJR
⋅(x − )=
( −− ) +)⋅ += + ( =− +)⋅ += = √
x= ⋅(x − )=
(
x= (x + )⋅( − x ) =
− )⋅ + + =−
=
+
− −
=− = −
− −
− −
x = −= x =−
=
=
=
=
=
=
=
=
⋅ + + == x −
=
=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x =−
=− − =− − += −
x =−
=− − =
x ==−− ⋅(x − )=
x =−
+
+
x = x = xx == x = a = a = aa == va = =
( ) (( )) (( ( ))) (( )) ) ( ) ( ) ( ) (( ))
−
v =
−
vv ==
−−
= under H:v ind på bagflappen. ( Cosinusrelationerne ) −
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
⋅(x − )=
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab
x= ⋅(x − )=
Kvadratsætningerne (a + b) = a + b + ab
x= (x⋅(x + −)⋅( )= −x ) = x= (x + )⋅( − x ) =
x = − eller x = x = − eller x =
⋅ +
)⋅ + + − x = ⋅ + x− = −x + = + = x−
= − = −− − − + − =− −
x− = − −
x = − eller x =
⋅ +
)⋅ +
−
=
=− + − =− − + = −− = − = −− − − − + + −= −= − −
x− = − −
x = − eller x =
=
−
√
− − − −
(x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) =
=
(
( −+ − )⋅ += − − ( − = − +√ = −
−
x= (x⋅(x + − )⋅( )= −x ) =
x == − eller x =
+ ⋅√
(a (a+− +b) b) ab b) === aaa+++bbb++− ab ab (a (a (a− + b)
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
(a (a−−b) b) == aa ++bb −− ab ab (a + b)(a − b) = a − b
x == − eller x =
b) −=b)a =+ ba−− b ab (a+−b)(a (a
(x + )⋅( − x ) =
(a (a++b)(a b)(a−−b) b) == aa −−bb
x = − eller x = (x + )⋅( − x ) =
(a + b)(a − b) = a − b
− + + ⋅√
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+ ⋅ −
=
+ = + ⋅ − = + = + ⋅ − =
( −− )+ + == − + ( −− )+ + == − +
= +
+ =
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
Kernestof Mat 2 hf
=
=
x = − eller x = x = − eller x =
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
(
)⋅ +
=
=
+
+
(
−
)⋅ +
( −+ − )⋅ += − − ( − = − +√ = −
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
−
− +
√
=−
=
− − − −
=− + − =− − + = −− = − = −− − − − + + −= −= − −
x− = − −
= − = −− − − + − =− −
x− = − − − −
=− = −
− −
− −
x = −= x =−
⋅ + + == x −
=
+
=
=
=
=
=
=
=
=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x =−
=− − =− − += −
x =−
=− − =
x ==−− ⋅(x − )=
x =− ⋅(x − )=
x= ⋅(x − )= x= (x⋅(x + − )⋅( )= −x ) = x= (x + )⋅( − x ) =
x == − eller x = (x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) = x = − eller x = x = − eller x =
( ) = = a(A ) b(B) c(C ) (( )) =⋅ =c − ⋅b⋅c a =b + (A) c(C ) b(B) a(A ) (( ( ))) =⋅ =c − ⋅b⋅c a =b + (A) c b a ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( = b a + c − ⋅b a⋅c ⋅ (B) ) ) a =b (A) (A ) (B) (C ) c = a + b − ⋅a⋅b⋅ (C ) a b c = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb − a = a =
aa ==
−
= va =
−
−−
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
−
c = abagflappen. +b (C ) b på c − ⋅a⋅b c⋅ (B) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A ) (B) (C ) c =(A) a= + b − ⋅a⋅b⋅ (C ) = = Kvadratsætningerne ⋅b⋅c a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne (a + b)
⋅b⋅c b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
= a + b + ab
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a a(A ) = = a (A ) (A )
x =
Kvadratsætningerne (a + b) = a + b + ab
x =
xx ==
x =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( b(B)))=
a = a =
aa ==
(a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a b) c= −a ⋅+bb − ab (a −
=
−
b
=−
c (C ) c
=
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C )
=
a(A ) b(B) a =b = + c − ⋅b⋅= c⋅ a(A ) b(B) a =b = + c − ⋅b⋅= c⋅ a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
c(C ) (A)
b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ a
(B) (A)
b a + c − ⋅b a⋅c ⋅ a =b c = a + b − ⋅a⋅b⋅ a = b + c − ⋅b⋅c ⋅
(B) (A) (C ) (A)
−
= va =
v = a vv == =−− (A ) a
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b (B) (B)
c(C ) (A)
(A )
(B)
=
(a + b)
(C )
=
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
b + c =(A) a =+ b −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) = a + c − b b (B )= a +⋅ba⋅− c cb (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
= a + b + ab
(a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − a + b − ab (a − b)
a = b + c= ⋅c+ ⋅ ab (A) (a− + b) = −a ⋅+bb − ab (a
(a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a
b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
(a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
(A) = (A) =
(B )=
(C ) = (C ) =
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a a(A ) = = a
x =
x =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c
(A ) (B) (C ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) =c − ⋅b⋅c =⋅ a =b + (A) a(A ) b(B) c(C ) (( ( ))) =c − ⋅b⋅c =⋅ a =b + (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b a + c − ⋅b a⋅c ⋅ (B) ))= (C ) a =b (A) (A ) (B) c = a + b − ⋅a⋅b⋅ (C ) a b c = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb − a = a =
(A) = (A) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
−
= va =
−
−−
−
c = abagflappen. +b (C b på c − ⋅a⋅b c⋅ (B)) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A )
(B)
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c =
(a + b)
=
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a (a − b) c= −a ⋅+bb − ab
a = b + c= ⋅c + ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab b) == aa ++bb −− ab ab (a (a−−b) b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(B )= (B )=
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
(C ) =
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
−
+
= − = +
+ =
= +
=
−
+
= − = +
+ =
= +
=
−
( − ) +
=
(B)
=
(C )
= a + b + ab
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a b) c= −a ⋅+bb − ab (a −
a = b + c= ⋅c+ ⋅ ab (A) (a− + b) = −a ⋅+bb − ab (a (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
( − ) +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
(C )
(C )
(
(A) =
(B )= (B )=
(C ) = (C ) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
)⋅ +
−
=
+√ = − − )⋅ +
−
(
= + √ −= −= − −
−
(
−
)⋅ +
(
−
)⋅ +
+
=
⋅ +
+
=
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
=
+ ⋅ −
=
+
−
=
−
=
+ ⋅ −
=
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
=
−
−
± ′
− x +==− =− − =− −
x =−
x ==−−
+
= − = + = − = +
(
)⋅ +
−
(
′
−
+√ = − − )⋅ +
=
= + √ −= −= − −
− += − = − − − + =− − − − = =− − −= − − − + −= −
=−
x ==−−
− − − += −− = − − −
x = −= ⋅(x − )=
x =−
x− −= − = −−
⋅(x − )=
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
−
=
−
x⋅= (x − )=
− −
= +
+ =
= +
= =
−
)⋅ +
(
−
)⋅ +
⋅ + ⋅ +
= =
+ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
+ −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x += − − =− − + −=x −= x − − x =− + =−− = −= − − + − x +==− =− − =− −
x =−
x ==−−
⋅(x − )=
=−
x =−
⋅(x − )=
x(x = + − )⋅( eller − x ) x= =
±∫
x⋅= (x − )= x⋅= (x − )=
x = − eller x =
(x + )⋅( − x ) =
x = − eller x =
x= (x + )⋅( − x ) = x= x = − eller x = (x + )⋅( − x ) =
x(x = + − )⋅( eller − x ) x= =
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a a(A ) = = a
x =
x =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b
x = − eller x =
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c
x = − eller x =
(A ) (B) (C ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) =c − ⋅b⋅= a =b + c ⋅ (A) a = a(A ) b(B) c(C ) (( ( ))) =c − ⋅b⋅= aa == a =b + c ⋅ (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) = va = b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b (B) ))= (C ) a =a b + c − ⋅a b⋅c ⋅ (A) (A ) (B) c = a + b − ⋅a⋅b⋅ (C ) c a b = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb −
+ ⋅√
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+
−
=
+
−
=
+
−
=
+
−
=
−
+ ⋅√
−
+ ⋅ −
=
−−
− + ⋅+√
+ +⋅ −= =
+ ⋅√
−
− + ⋅+√
c = abagflappen. +c b − ⋅a⋅c b⋅ (B) (C ) b på H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A )
=
(B)
=
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
+ c ⋅− c =(A) a =+ b b − a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
(C )
= a + b + ab
(a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a (a − b) c= −a ⋅+bb − ab
a = b + c= ⋅c+ ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b (a (a++b)(a b)(a−−b) b) == aa −−bb c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
−
(B )=
(C ) =
−
+ +⋅ −= =
+ ⋅√
+
= − = +
( − ) + +
= − = +
( − ) +
+ =
= +
+ =
= +
=
− + ⋅+√
=
− + ⋅+√
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √ (
)⋅ +
−
=
+√ = − − )⋅ +
−
(
= + √ −= −= − −
−
(
−
)⋅ +
(
−
)⋅ +
− += − = − − − + =− − − − = =− − −= − − − + −= −
=
+
=
⋅ +
+
=
−
−
x− = −= −
− x +==− =− − =− −
x =−
x ==−−
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+ ⋅ −
=
+ +⋅ −= = + +⋅ −= =
= − = +
+ =
= +
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
=
( − ) +
=
=
( − ) +
=
=
=
=
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
−
(
+
(
−
= − = +
)⋅ +
−
−
+√ = − − )⋅ +
=
= + √ −= −= − −
− += − = − − − + =− − − − = =− − + −= − −= − − − − − − += −− = − − −
x = −=
=−
x− −= − = −− =
−
x ==−− ⋅(x − )=
+
=
=− + =−− = −= − − +
−
−
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
⋅ +
+= − − =− − + −=x −= x − − x
x− −= − = −−
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x + −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x
− − − += −− = − − −
x = −=
−
(B )=
(C ) =
−
x =−
− −
= =
(
−
)⋅ +
(
−
)⋅ +
⋅ + ⋅ +
= =
+ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=
=
=
=
=
=
=
=
+ −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x += − − =− − + −=x −= x − − x =− −− + =−= + =− −
x− = −= −
− x +==− =− − =− −
x =−
x ==−−
=−
⋅(x − )=
x ==−− x⋅= (x − )=
⋅(x − )=
x =−
x⋅= (x − )= ⋅(x − )=
x= (x + )⋅( − x ) =
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
x⋅= (x − )= x⋅= (x − )= (x + )⋅( − x ) =
x(x = + − )⋅( eller − x ) x= =
x= (x + )⋅( − x ) =
x = − eller x =
x= x = − eller x = (x + )⋅( − x ) =
x = − eller x = x(x = + − )⋅( eller − x ) x= = a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a a(A ) = = a
x = x = xx == x =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B) = = b
= − = +
+ =
= +
=
−
+
= − = +
+ =
= +
=
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c
x = − eller x = x = − eller x =
(A ) (B) (C ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) a =b = + c − ⋅b⋅= c ⋅ (A) a = a(A ) b(B) c(C ) (( ( ))) aa == a =b = + c − ⋅b⋅= c ⋅ (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) = va = b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b ))= (C ) a =a b + c − ⋅a b⋅c ⋅ (B) (A) (A ) (B) c = a + b − ⋅a⋅b⋅ (C ) c a b = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb −
Kernestof Mat 1 hfKernestof Mat 2 hf
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √ (
−
−
+√ = − − )⋅ +
(
a =
)⋅ +
=
= + √ −= −= − −
−
a =
(a + b)
+
( − ) +
x ==−−
x= (x + )⋅( − x ) = x= x = − eller x = (x + )⋅( − x ) =
∫
(
x− = −= −
x⋅= (x − )=
∫
∫
+ =
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
−
=− + =−− = −= − − +
−
x− = −= −
+
−
( − ) +
′
+= − − =− − + −=x −= x − − x
−
−
′
+ −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x
− − − += −− = − − −
x− −= − = −−
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(x + )⋅( − x ) =
∫ ±
x =
xx ==
(x + )⋅( − x ) =
(C ) =
−
+ ⋅√
− + ⋅+√
( − ) + ′ ′
± − += − = − − − + =− − − − = =− − + −= − −= − − −
x = −=
(A) =
⋅ +
x= x = − eller x = (x + )⋅( − x ) =
(C ) =
+ ⋅√
− + ⋅+√
−
(A )
(a + b)
Kvadratsætningerne (a + b) = a + b + ab
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
′
−
−−
c = abagflappen. (C ) b på +b c − ⋅a⋅b c⋅ (B) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(C ) =
−
( − ) +
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind på c = abagflappen. + b − ⋅a⋅b⋅ (C ) (A) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
x =
xx ==
aa ==
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c (C c ) c
(C ) (B) (A ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) =⋅ a =b = + c − ⋅b⋅c (A) a = a(A ) b(B) c(C ) (( ( ))) =⋅ aa == a =b = + c − ⋅b⋅c (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) = va = b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b a + c − ⋅b a⋅c ⋅ (B) ))= (C ) a =b (A) (A ) (B) c = a + b − ⋅a⋅b⋅ (C ) c b a = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb −
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(B )=
b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
(B )=
x = a =
c b⋅ (B) (C ) b på = abagflappen. +b c − ⋅a⋅c H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a
b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b
(B )=
x = xx ==
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c Kvadratsætningerne (a + b) = a + b + ab
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a
x =
c (A) (B)
= (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind b = ++ cb −
a = b + c= ⋅c + ⋅ ab (A) (a− + b) = −a ⋅+bb − ab (a
− + ⋅+√
Denne udgave af Kernestof Mat2 er målrettet matematikundervisningen på hf-uddannelsen ifølge læreplanerne fra 2024. Bogen dækker anden del af kernestoffet på B- og A-niveau.
(C )
(
−
)⋅ +
(
−
)⋅ +
− += − = − − − + =− − − − = =− − + −= − −= − − −
−
= =
+ +
=
=
=
=
=
=
=
=
+ −= −= − − + −=x x=− x − − x += − − =− − + −=x −= x − − x
x− −= − = −− −
⋅ +
+ −=x −= x − − x + = x−
− − − += −− = − − −
x = −= =
⋅ +
⋅ + + = = = x− ⋅ + + = = + −=x x=− x − − x +
=− + =−− = −= − − +
− −
x− = −= −
− x +==− =− − =− −
x =−
x ==−−
Bogen har en let og overskuelig opbygning. Hvert opslag er et afgrænset, øvelsesbaseret læringsforløb. En facitliste til alle øvelser findes bagerst i bogen.
−
−
−−
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
= −
− =
+ +⋅ −= =
+
−
=
−
c b − ⋅a⋅c b⋅ (B) (C ) b på = abagflappen. +c H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A )
(B)
(C )
c =(A) a= + b − ⋅a⋅b⋅ ⋅b⋅c
(C )
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c =
(a + b)
=
= a + b + ab
c =(A) a =+ b b + −c ⋅− a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa ++bb⋅c++⋅ ab ab(A) a (a − b) c= −a ⋅+bb − ab
a = b + c= ⋅c + ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
b +c − a ⋅b⋅c b +c − a (A) = ⋅b⋅c (A) =
(B )= (B )=
(C ) = (C ) =
− + ⋅+√
=−
(C ) −
+
= − = +
( − ) + −
+
= − = +
( − ) +
+ =
= +
=
+ =
= +
=
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + = ( − )⋅ + = = √ (
−
−
+√ = − − )⋅ +
(
−
)⋅ +
=
= + √ −= −= − −
(
−
)⋅ +
(
−
)⋅ +
− += − = − − − + =− − − − = =− − −= − − − + −= −
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
−
=
=
+ +
=
=
=
=
=
=
=
=
x ==−−
+ −=x −= x − − x + = x−
⋅(x − )=
+= − − =− − + −=x −= x − − x
x− −= − = −− −
=
⋅ +
+ −= −= − − + −=x x=− x − − x
− − − += −− = − − −
x = −=
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
⋅ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x
=− + =−− = −= − − +
− −
x− = −= −
− x +==− =− − =− −
x =−
x ==−−
x =−
=−
x ==−− ⋅(x − )=
x =−
⋅(x − )=
⋅(x − )=
x⋅= (x − )= x⋅= (x − )= (x + )⋅( − x ) = x= (x + )⋅( − x ) = x= x = − eller x = (x + )⋅( − x ) =
x(x = + − )⋅( eller − x ) x= = x = − eller x =
x⋅= (x − )=
x = − eller x =
Lindhardt og Ringhof
x⋅= (x − )= (x + )⋅( − x ) = x= (x + )⋅( − x ) =
Per Gregersen og Majken Sabina Skov
x= x = − eller x = (x + )⋅( − x ) =
x = x = xx == x = a = a = aa == va = =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( ))
Praxis
−
v =
−
vv ==
−−
= under H:v ind på bagflappen. ( Cosinusrelationerne ) −
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab + ⋅√
Kvadratsætningerne (a + b) = a + b + ab (a− + b) (a
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+
−
=
+
−
=
+
−
=
+
−
=
x =
((a (a a +− +b) b) ab b) === aaa+++bbb++− ab ab
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
((a a −−b) b) == aa ++bb −− ab ab (a + b)(a − b) = a − b
− + + ⋅√
(a b) = a + b − ab (a +−b)(a − b) = a − b b)(a−−b) b) == aa −−bb ((a a ++b)(a
+ ⋅ −
=
x = + = + ⋅ − =
xx == x =
+ = + ⋅ − =
a =
( −− )+ + == − +
= +
+ =
+ =
= +
=
a =
(a + b)(a − b) = a − b ( −− )+ + == − +
aa ==
=
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
=
=
+
+
−
)⋅ +
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
(
−
− )⋅ + + =−
√
=
=− + − =− − + = −− = − = −− − − + −= −= − − = − = −− − − + − =− −
x− = − − − −
=− = −
− −
− −
x = −= x =−
⋅ + + == x −
=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x =−
=− − =− − += −
x =−
va = = =
=
=
=
=
=
=
=
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( ))
−
v =
−
vv ==
−−
x(x = + − )⋅( eller − x ) x= =
= under H:v ind på bagflappen. ( Cosinusrelationerne ) −
(
( −+ − )⋅ += − − ( − = − +√ = − − − − −
x− = − −
+
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab + ⋅√
Kvadratsætningerne (a + b) = a + b + ab (a (a+− +b) b) ab b) === aaa+++bbb++− ab ab (a− + b) (a
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
b) == aa ++bb −− ab ab (a (a−−b) (a + b)(a − b) = a − b
− + + ⋅√
(a − b) = a + b − ab (a + b)(a − b) = a − b (a (a++b)(a b)(a−−b) b) == aa −−bb
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+ ⋅ −
=
+ = + ⋅ − = + = + ⋅ − =
( −− + ) + == − +
=− − =
= +
+ =
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
=
(a + b)(a − b) = a − b x ==−−
( −− + ) + == − +
x =−
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
=
⋅(x − )=
⋅(x − )=
x= ⋅(x − )=
+
)⋅ + + − x = ⋅ + x− = −x + = + = x−
√
=
=− + − =− − + = −− = − = −− − − − + + −= −= − −
x− = − −
= − = −− − − + − =− −
x− = − −
x = − eller x =
+
)⋅ +
−
− )⋅ + + =−
− − − −
(x + )⋅( − x ) =
x = − eller x =
=
−
(
x= (x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) =
=
(
( −+ − )⋅ += − − ( − = − +√ = −
−
x= (x⋅(x + − )⋅( )= −x ) =
x == − eller x =
− −
=− = −
− −
− −
x = −= x =−
⋅ + + == x −
=
+
=
=
=
=
=
=
=
=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x =−
=− − =− − += −
Per Gregersen og Henrik Bindesbøll Nørregaard
x =−
=− − =
x ==−− ⋅(x − )=
x =− ⋅(x − )=
x= ⋅(x − )= x= (x⋅(x + − )⋅( )= −x ) = x= (x + )⋅( − x ) =
x == − eller x = (x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) = x = − eller x = x = − eller x =
fp_k=VTUJUTJTMSSJUSTJR
x =
x =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b b(B)
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C )
a
b
c
= =
= =
x = − eller x = x = − eller x =
(C ) (B) (A ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) =⋅ a =b = + c − ⋅b⋅c (A) a(A ) b(B) c(C ) (( ( ))) =⋅ a =b = + c − ⋅b⋅c (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b a + c − ⋅a ))= (C ) a =b b⋅c ⋅ (B) (A) (B) (A ) c = a + b − ⋅a⋅b⋅ (C ) c b a = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb −
Flere end 130 QR-koder undervejs i bogen linker til screencasts med uddybende forklaringer på begreber, formler, sætninger og beviser. Videoerne er også velegnede til flipped classroom-undervisning og til eksamensforberedelse.
a = a =
aa ==
= va =
−
−
−−
−
c = abagflappen. (C ) b på +b c − ⋅a⋅b c⋅ (B) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A ) (B) (C ) c =(A) a= + b − ⋅a⋅b⋅ (C ) = = Kvadratsætningerne ⋅b⋅c a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne (a + b)
⋅b⋅c b − + c ⋅− c =(A) a =+ b a⋅ab⋅ ⋅b⋅c a +c − b (B )= b + c − a (A) = a +⋅ca⋅−c b ca (B )= b +⋅cb⋅− (A) = ⋅a⋅c a +c − b (B )= b +⋅cb⋅−c a (A) = ⋅a⋅c cc a +⋅b⋅− (C ) =a + c − b b (B )= a +⋅ba⋅− c bc (C ) = a +⋅ca⋅− (B )= ⋅a⋅b a + b −c (C ) = a +⋅ca⋅−c b (B )= ⋅a⋅b ⋅a⋅c a + b −c (C ) = a⋅− bc a +⋅b (C ) = bc a +⋅ba⋅− (C ) = ⋅a⋅b
= a + b + ab
Kvadratsætningerne (a + b) = a + b + ab (a (a++=b) b)b +== aa +⋅+bbb⋅c++⋅ ab ab(A) a c= − (a − b) a + b − ab
a = b + c= ⋅c + ⋅ ab (A) (a− + b) (a = −a ⋅+bb − ab (a (a−−b) b) == aa ++bb −− ab ab b a +− c b) − ⋅= a⋅ca⋅ − b(B) (a +=b)(a b(a = a + c=−a⋅a (B) − b) +⋅bc ⋅− ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a c = a + b − ⋅a⋅b⋅ (C )
c(a = a + b− b) − ⋅= a⋅ba⋅ − b(C ) + b)(a
(A) = (A) =
(B )= (B )=
(C ) = (C ) =
b +c − a ⋅b⋅c b +c − a ⋅b⋅c
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
− + ⋅+√
= −
− =
+ +⋅ −= =
+
−
=
−
+
= − = +
+ =
= +
=
+
= − = +
+ =
= +
=
(C ) ( − ) + −
KernestofKernestof Mat 1 hhxMat 2 hhxKernestof Mat 3 hhx ( − ) +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √ (
−
−
+√ = − − )⋅ +
(
−
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
)⋅ +
=
= + √ −= −= − −
a =
( ) (( )) (( ( ))) ( ( )) ) ( ) ( ) ( ) (( ))
−
−−
= under H:v ind på bagflappen. ( Cosinusrelationerne ) −
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen.
=
=
+
+
=
=
=
=
=
=
=
=
+ −=x −= x − − x + = x−
Alle kapitler afsluttes med opgaver, der følger den faglige progression. På bogens website er der facitlister til alle bogens opgaver.
=− −− + =−= + =− −
− −
x− = −= −
− x +==− =− − =− −
x =−
x ==−−
=−
⋅(x − )=
Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b) = a + b + ab + ⋅√
−
=
+ ⋅√
−
=
− + + ⋅√
= −
− =
= −
− =
Kvadratsætningerne (a + b) = a + b + ab (a (a+− +b) b) ab b) === aaa+++bbb++− ab ab (a− + b) (a
=
x ==−−
−
v =
⋅ +
⋅ + + = = = x− ⋅ + + = = + −=x x=− x − − x +
+= − − =− − + −=x −= x − − x
−
x = a =
va = =
)⋅ +
+ −= −= − − + −=x x=− x − − x
−
x =
aa ==
vv ==
)⋅ +
−
− − − += −− = − − −
x− −= − = −−
x =
xx ==
−
(
− += − = − − − + =− − − − = =− − −= − − − + −= −
x = −=
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
(
⋅ +
= = a +b + − ab ab
(a (a−−b) b) == aa ++bb −− ab ab (a + b)(a − b) = a − b (a − b) = a + b − ab (a + b)(a − b) = a − b b)(a−−b) b) == aa −−bb (a (a++b)(a
− + + ⋅√
+ ⋅ −
=
+ ⋅ −
=
+ = + ⋅ − = + = + ⋅ − =
( −− + ) + == − +
= +
+ =
=
+ =
= +
=
+
−
=
+
−
=
+
−
=
+
−
=
( −− + ) + == − +
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
(
− )⋅ + + =−
+
=
+
=
(
−
)⋅ +
)⋅ += − − ( ( −+ − − = − +√ = −
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
−
√
=
− − − −
=− + − =− − + = −− = − = −− − − − + + −= −= − − = − = −− − − + − =− −
x− = − − − −
=− = −
− −
− −
x = −=
⋅ + + == x −
+
=
=
=
=
=
=
=
=
=
x⋅= (x − )=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x− = − −
Lindhardt og Ringhof
x =−
=− − =− − += −
x =−
=− − =
x =−
x =−
⋅(x − )=
(a + b)(a − b) = a − b
x ==−− ⋅(x − )=
x =− ⋅(x − )=
x⋅= (x − )= (x + )⋅( − x ) =
x= ⋅(x − )= x= (x⋅(x + − )⋅( )= −x ) = x= (x + )⋅( − x ) =
x == − eller x = (x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) =
Per Gregersen og Majken Sabina Skov
x = − eller x = x = − eller x =
x= (x + )⋅( − x ) =
Lindhardt og Ringhof
x= x = − eller x = (x + )⋅( − x ) =
x(x = − eller = +og )⋅(Henrik − x ) x=Bindesbøll Per Gregersen Nørregaard
Praxis
x = − eller x = x = − eller x =
Henrik Bindesbøll Nørregaard og Per Gregersen
flappen.
flappen. flappen. flappen.
gflappen. + ⋅√
−
=
+ ⋅ −
=
+
−
=
+ ⋅√
−
=
+ ⋅ −
=
+
−
=
− + + ⋅√
= −
− =
+ = + ⋅ − =
+
−
=
= −
− =
+ = + ⋅ − =
+
−
=
x =
− + + ⋅√
x = xx == x =
( ) (( )) a = (( ( ))) aa == ( ( )) ) va = = ( ) ( ) v = ( ) vv == (( )) = under H:v ind på bagflappen. ( Cosinusrelationerne ) a =
( −− )+ + == − + ( −− )+ + == − +
= +
+ =
+ =
= +
=
=
−
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
(
=
+
+
)⋅ +
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
√
=
− − − −
=− + − =− − + = −− = − = −− − − + −= −= − − − +
x− = − −
= − = −− − − + − =− −
x− = − − − −
=− = −
− −
− −
x = −= x =−
⋅ + + == x −
=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x =−
=− − =− − += −
x =−
=
=
=
=
=
=
=
=
−
−−
−
−
−
− )⋅ + + =−
=
(
( −+ − )⋅ += − − ( − = − +√ = −
+
Kernestof Mat 2 hf
2. udgave
KernestofKernestof Mat 1 htxMat 2 htxKernestof Mat 3 htx
Mellem bogens kapitler er der træningssider med enkle forklaringer og korte opgaver til vedligeholdelse af basale regnefærdigheder og repetition af grundlæggende begreber som regningsarternes hierarki, algebra, brøkregning med videre.
Kernestof Mat 2 hf
x =
xx ==
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A a )
Per Gregersen & Henrik Bindesbøll Nørregaard
x⋅= (x − )= x= (x + )⋅( − x ) =
x = − eller x =
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c (C ) c c
=
+= − − =− − + −=x −= x − − x
x ==−−
⋅(x − )=
x⋅= (x − )=
(x + )⋅( − x ) =
x(x = + − )⋅( eller − x ) x= =
x = − eller x = b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b
(A ) (B) (C ) ( ) (A ) (B) (C ) = = a(A ) b(B) c(C ) (( )) =c − ⋅b⋅= a =b + c ⋅ (A) a(A ) b(B) c(C ) (( ( ))) =c − ⋅b⋅= a =b + c ⋅ (A) a b c ( ( )) ) a = b + c − ⋅b⋅c ⋅ (A) b = a + c − ⋅a⋅c ⋅ (B) ( ) ( ) b = =b a ++ cc − − ⋅⋅b a⋅⋅cc⋅⋅ (A) (B) v = a (b ) c a vv == = (( b a⋅c ⋅ (B) ))= (C ) a =a b + c − ⋅b (A) (A ) (B) c = a + b − ⋅a⋅b⋅ (C ) a b c = a = b + c − ⋅b⋅c ⋅ (A) = (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne )= (C ) H:v ind b = ++ cb −
⋅ +
+ −= −= − − + −=x x=− x − − x
x ==−−
⋅(x − )=
x= (x + )⋅( − x ) =
x =
⋅ +
⋅ + + = = + = x− ⋅ + + = = + −=x x=− x − − x + −=x −= x − − x + = x−
− x +==− =− − =− −
x =−
x⋅= (x − )= (x + )⋅( − x ) =
x= x = − eller x = (x + )⋅( − x ) =
x =
+
=
)⋅ +
−
= +
x− −= − = −−
=
x− = −= −
⋅(x − )=
x⋅= (x − )=
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a
x =
xx ==
= +
−
=
+ =
− − x = −= − += −− = − − −
a =
aa ==
Kvadratsætningerne (a + b) = a + b + ab
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
(C ) =
x(x = + − )⋅( eller − x ) x= =
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind på c = abagflappen. + b − ⋅a⋅b⋅ (C ) (A) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(C ) =
x= (x + )⋅( − x ) = x= x = − eller x = (x + )⋅( − x ) =
xx ==
= +
+ =
x ==−−
⋅(x − )=
x(x = + − )⋅( eller − x ) x= =
±∫
+ =
− x +==− =− − =− −
⋅(x − )=
x= x = − eller x = (x + )⋅( − x ) =
∫ ±
c (C ) c (C ) c (C ) (C ) c c(C ) c(C ) c c(C ) c (C ) (C )
+ +⋅ −= =
(
x =−
x ==−−
x= (x + )⋅( − x ) =
b = (B) b = (B) b = (B) (B) = b b(B) = b(B) = = b (B) b = = b (B) (B)
=
+ +⋅ −= =
(
x−= −= −
x⋅= (x − )= (x + )⋅( − x ) =
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a (A ) (A )
=
+ ⋅ −
= − = +
)⋅ +
−
(
′
=−
x ==−− ⋅(x − )=
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
+ ⋅ −
= − = +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
−
=− + =−− = −= − − +
−
x− = −= −
+
( − ) +
′
+= − − =− − + −=x −= x − − x
−
−
′
+ −=x −= x − − x + = x− + −= −= − − + −=x x=− x − − x
− − − += −− = − − −
x− −= − = −−
+
( − ) + ′ ′
± − += − = − − − + =− − − − = =− − + −= − −= − − −
x = −=
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
= +
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ (( −− ) +)⋅ += + = + ( − )⋅ += = = √
′
=
(( −− ) +)⋅ += + ( =− +)⋅ += = = √ ( −− ) +)⋅ += + = + ( − )⋅ += ( = = √
(A) =
+
( − ) +
= (A ) under c på =a abagflappen. + cb − − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind b = +
c = abagflappen. +c b − ⋅a⋅c b⋅ (C b på (B)) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a
Kvadratsætningerne a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne ⋅b⋅c
a + b −c ⋅a⋅b a + b −c ⋅a⋅b
(C ) =
a = (A ) a = (A ) a = (A ) (A ) = a a(A ) = a(A ) = = a (A ) a = = a (A ) (A )
x = xx ==
aa ==
(C )
a +c − b ⋅a⋅c a +c − b ⋅a⋅c
(B )=
(C ) =
a(A ) b(B) =c − ⋅b⋅= a =b + c⋅ a(A ) b(B) =c − ⋅b⋅= a =b + c⋅ a b a = b + c − ⋅b⋅c ⋅ b = a + c − ⋅a⋅c ⋅
x =
c (A) (B)
= (A ) under c på =a abagflappen. − ⋅⋅a a⋅⋅cb⋅⋅ (B) (C ) ( (B)Cosinusrelationerne ) (C ) H:v ind b = ++ cb −
c b − ⋅a⋅c b⋅ (C b på = abagflappen. +c (B)) H: ind Cosinusrelationerne (A ) under(B) (C ) = = Kvadratsætningerne b på = abagflappen. + c − ⋅a⋅c ⋅ (B) H: H: ind ind under Cosinusrelationerne på bagflappen. a under b Cosinusrelationerne c b +c − a (A ) (B) (C ) c =(A) a= + b − ⋅a⋅b⋅ (C ) = = Kvadratsætningerne ⋅b⋅c a under b Cosinusrelationerne c b +c − a H: ind c på =(A) abagflappen. + b − ⋅a⋅b⋅ (C ) = Kvadratsætningerne Kvadratsætningerne
+ =
KernestofKernestof Mat 1 stxMat 2 stxKernestof Mat 3 stx ( − ) +
(B )=
(C ) =
−
+ ⋅√
− + ⋅+√
(C )
x =
(C ) =
+ ⋅√
− + ⋅+√
Bogens website: prx.dk/kernestof
Praxis
H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne H: H: ind ind under under Cosinusrelationerne Cosinusrelationerne på på bagflappen. bagflappen. Kvadratsætningerne H: ind under Cosinusrelationerne på bagflappen. Kvadratsætningerne Kvadratsætningerne (a + b)
= a + b + ab
+ ⋅√
Kvadratsætningerne (a + b) = a + b + ab (a +b) b) ab (a+− b) === aaa+++bbb++− ab ab (a− + b) (a
+ ⋅√
= = a +b + − ab ab
− + + ⋅√
(a (a−−b) b) == aa ++bb −− ab ab (a + b)(a − b) = a − b
− + + ⋅√
b) = a + b − ab (a+−b)(a − b) = a − b (a b)(a−−b) b) == aa −−bb (a (a++b)(a
=− − =
−
=
−
=
= −
− =
= −
− =
+ ⋅ −
=
+ ⋅ −
=
+ = + ⋅ − = + = + ⋅ − =
( −− )+ + == − +
= +
+ =
+ =
= +
+
−
=
+
−
=
+
−
=
+
−
=
=
(a + b)(a − b) = a − b x ==−−
( −− )+ + == − +
x =−
( − )⋅ + ( − )⋅ + ( − ) + = =+ = + = = √
⋅ +
( −− ) +)⋅ += + ( =− +)⋅ += = = √
⋅ +
=
⋅(x − )=
⋅(x − )=
x= ⋅(x − )= x= (x⋅(x + − )⋅( )= −x ) = x= (x + )⋅( − x ) =
x == − eller x = (x + )⋅( − x ) =
− )⋅ + + =−
=
+
+
−
)⋅ +
−
)⋅ + + − x = ⋅ + x− = −x + = + = x−
(
−
√
=
=− + − =− − + = −− = − = −− − − − + + −= −= − −
x− = − −
x = − eller x =
x− = − −
x = − eller x =
=
(
)⋅ += − − ( ( −+ − − = − +√ = − − − − −
x = − eller x = (x + )⋅( − x ) =
= − = −− − − + − =− − − −
=− = −
− −
− −
x = −= x =−
⋅ + + == x −
=
+
=
=
=
=
=
=
=
=
+ =− + −x = x − −x + = x− + − =− − + =− + + −=x x=− x − − x =− + − =− − + + −=x −= x − − x =− =− + −= −= − −
x =−
=− − =− − += −
x =−
=− − =
Lindhardt og Ringhof
x ==−− ⋅(x − )=
x =− ⋅(x − )=
x= ⋅(x − )= x= (x⋅(x + − )⋅( )= −x ) = x= (x + )⋅( − x ) =
x == − eller x = (x + )⋅( − x ) =
x = − eller x = (x + )⋅( − x ) = x = − eller x = x = − eller x =
Per Gregersen og Henrik Bindesbøll Nørregaard
Lindhardt og Ringhof
Per Gregersen og Henrik Bindesbøll Nørregaard
Praxis
Henrik Bindesbøll Nørregaard og Per Gregersen
P R A X I S . D K
2. udgave
Per Gregersen & Henrik Bindesbøll Nørregaard
ISBN 978-87-2901-866-7
prx.dk/kernestof
9788729018667_omslag.indd 1
9 788729 018667
26.05.2025 12.56