Mathematics Question and Solution
Calculating the required angle.
Notice.
The ascribed plane shape is a square, let it's side length be x.
Let the longest side length of the inscribed red lengths triangle be 1 unit.
a = 180-45-57
a = 180-102
a = 78°
(1/sin78) = (b/sin45)
b = 0.72290396731 units.
(1/sin78) = (c/sin57)
c = 0.85740696733 units.
d²+x² = 0.85740696733²
d = √(0.85740696733²-x²) units.
cosy = x/0.85740696733 --- (1).
siny = √(0.85740696733²-x²)/0.85740696733 --- (2).
cos(45-y) = x/1
cos45cosy+sin45siny = x
½√(2)(cosy+siny) = x
√(2)(cosy+siny) = 2x --- (3).
Calculating x.
Substituting (1) and (2) in (3).
√(2)((x/0.85740696733)+(√(0.85740696733²-x²)/0.85740696733)
) = 2x
√(2)((x)+(√(0.85740696733²-x²)) = 1.71481393466x
x+√(0.85740696733²-x²) = 1.21255656167x
√(0.85740696733²-x²) = 0.21255656167x
0.85740696733²-x² = 0.04518029191x²
1.04518029191x² = 0.73514670763
x² = 0.73514670763/1.04518029191
x = √(0.73514670763/1.04518029191)
x = 0.83867056795 units.
Recall.
At (1).
cosy = x/0.85740696733
And x = 0.83867056795 units.
cosy = 0.83867056795/0.85740696733
y = acos(0.83867056795/0.85740696733)
y = 12°
e = 45+y
e = 45+12
e = 57°
Therefore, the required angle is, let it be f.
f = 180-e-45
f = 180-57-45
f = 180-102
f = 78°
Again, f is the required angle.
