Expression 19: "A" equals StartFraction, left parenthesis, "a" plus "b" , right parenthesis Over 2 , EndFraction times "h"A=a+b2·h
19
Expression 20: 2 "A" equals "h" left parenthesis, "a" plus "b" , right parenthesis2A=ha+b
20
Expression 21: 2 "A" equals "h" "a" plus "h" "b"2A=ha+hb
21
Expression 22: 2 "A" minus "h" "b" equals "h" "a"2A−hb=ha
22
Expression 23: "a" equals StartFraction, 2 "A" minus "h" "b" Over "h" , EndFractiona=2A−hbh
23
Now we substitute a into our equation above and solve for h
24
Expression 25: left parenthesis, "a" minus "b" , right parenthesis tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis equals 2 "h"a−btanθ1=2h
25
Expression 26: left parenthesis, StartFraction, 2 "A" minus "h" "b" Over "h" , EndFraction minus "b" , right parenthesis tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis equals 2 "h"2A−hbh−btanθ1=2h
26
Expression 27: left parenthesis, StartFraction, 2 "A" minus 2 "h" "b" Over "h" , EndFraction , right parenthesis tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis equals 2 "h"2A−2hbhtanθ1=2h
27
Expression 28: left parenthesis, 2 "A" minus 2 "h" "b" , right parenthesis tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis equals 2 "h" squared2A−2hbtanθ1=2h2
28
Expression 29: left parenthesis, "A" minus "h" "b" , right parenthesis tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis equals "h" squaredA−hbtanθ1=h2
29
Expression 30: "A" tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis minus "h" "b" tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis equals "h" squaredAtanθ1−hbtanθ1=h2
30
Expression 31: 0 equals "h" squared plus "h" "b" tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis minus "A" tangent left parenthesis, theta Subscript, 1 , Baseline , right parenthesis0=h2+hbtanθ1−Atanθ1
31
Quadratic formula time! fun :)
a = 1
b = btan(θ)
c = -Atan(θ)
32
Expression 33: "h" equals StartFraction, negative "b" tan left parenthesis, theta Subscript, 1 , Baseline , right parenthesis plus StartRoot, "b" squared tan squared left parenthesis, theta Subscript, 1 , Baseline , right parenthesis plus 4 "A" tan left parenthesis, theta Subscript, 1 , Baseline , right parenthesis , EndRoot Over 2 , EndFractionh=−btanθ1+b2tan2θ1+4Atanθ12
