Mathematics Question and Solution
Let r be the radius of the quarter circle.
Let x be the interior angle facing 1 cm.
a = (x+45)°
a is alternate to y, the angle formed by the meet point of r and 2 cm at the circumference of the circle.
It implies;
y = (x+45)°
sinx = 1/r --- (1).
cosy = 2/r
cos(x+45) = 2/r
cosxcos45-sinxsin45 = 2/r
½√(2)(cosx-sinx) = 2/r --- (2).
Notice.
At (1).
Sinx = 1/r
Where;
1 cm = opposite.
r cm = hypotenuse.
Therefore, let b be the adjacent.
It implies;
b²+1² = r²
b² = r²-1
b = √(r²-1) cm.
It implies;
cosx = b/r
cosx = √(r²-1)/r --- (3).
Substituting (1) and (3) in (2).
½√(2)((√(r²-1)/r)-(1/r)) = 2/r
√(2)((√(r²-1)/r)-(1/r)) = 4/r
√(2)(√(r²-1)-1)/r = 4/r
√(2)(√(r²-1)-1) = 4
√(r²-1)-1 = 2√(2)
√(r²-1) = 2√(2)+1
r²-1 = (2√(2)+1)²
r² = (2√(2)+1)²+1
r² = 15.6568542495
r = √(15.6568542495)
r = 3.95687430297 cm.
Checking Accuracy.
asin(1/r)+asin(2/r) must equal 45°
And r = 3.95687430297 cm.
Therefore;
asin(1/3.95687430297)+asin(2/3.95687430297) = 45°
