Geometri

Trekanter

$drawing( 320, 240, 0, 10, 0, 10, triangle(1,1, 9,1, 9,9), locate(0.8,1, A), locate(9,10, B), locate(9,1, C), locate(9.1,5.1, a), locate(4.9,1, b), locate(4.5,5.5, c) )$

For alle trekanter gælder:

\[c^2 = a^2 +b^2 -2ab \cdot cos(C)\]

\[\frac{ a }{ sin(A) } = \frac{ b }{ sin(B) } = \frac{ c }{ sin(C) }\]

For retviklede trekanter gælder:

\[c^2 = a^2 +b^2\]

\[sin(V) = \frac{ mod }{ hyp }\]

\[cos(V) = \frac{ hos }{ hyp }\]

\[tan(V) = \frac{ mod }{ hos }\]

Bevis

Figur 1
$4*(ab/2) +a^2 +b^2 = (a+b)^2 = 4*(ab/2) +c^2$
$4*(ab/2) +a^2 +b^2 = cross((a+b)^2) = 4*(ab/2) +c^2$
$4*(ab/2) +a^2 +b^2 = 4*(ab/2) +c^2$
$cross(4*(ab/2)) +a^2 +b^2 = cross(4*(ab/2)) +c^2$
$+a^2 +b^2 = +c^2$

Bevis

$drawing( 640, 240, 0, 20, 0, 10, triangle(1,1, 6,1, 6,6), locate(0.8,1, A), locate(6,7, B), locate(6,1, C), locate(6.1,4.1, sin(A)), locate(2.9,1, cos(A)), locate(2.5,3.5, 1), triangle(11,1, 19,1, 19,9), locate(10.8,1, A), locate(19,10, B), locate(19,1, C), locate(19.1,5.1, a), locate(14.9,1, b), locate(14.5,5.5, c) )$

$k = c/1 = c$

$c*sin(A) = a$
$sin(A) = a/c$

$c*cos(A) = b$
$cos(A) = b/c$

Bevis

Definition: $tan(A) = sin(A)/cos(A)$

$tan(A) = sin(A)/cos(A) = (a/c)/(b/c) = (a/c)*(c/b) = (a*cross(c))/(cross(c)*b) = a/b$
$tan(A) = a/b$

Bevis

$drawing( 640, 240, 0, 20, 0, 10, triangle(1,1, 14,1, 9,9), triangle(1,1, 9,1, 9,9), locate(0.8,1, A), locate(9,10, B), locate(14,1, C), locate(9,1, D), locate(12.1,5.1, a), locate(9.1,5.1, h), locate(4.9,1, b-x), locate(10.9,1, x), locate(4.5,5.5, c) )$

$cos(C) = x/a$ <=> $a*cos(C) = x$

$h^2 +x^2 = a^2$ <=> $h^2 = a^2 -x^2$
$h^2 +(b-x)^2 = c^2$ <=> $h^2 = c^2 -(b-x)^2$
$c^2 -(b-x)^2 = h^2 = a^2 -x^2$

$c^2 -(b-x)^2 = a^2 -x^2$
$c^2 = a^2 -x^2 +(b-x)^2$
$c^2 = a^2 -x^2 +(b-x)*(b-x)$
$c^2 = a^2 -x^2 +b^2 +x^2 -2bx$
$c^2 = a^2 +b^2 -2bx$
$c^2 = a^2 +b^2 -2b * a*cos(C)$
$c^2 = a^2 +b^2 -2ab*cos(C)$

Beviser for Pythagoras