Mathematics Question and Solution
Let the square side be 2 units.
Area Square is;
2² = 4 square units.
tana = 2/1
a = atan(2)
b = 180-2a
b = (180-2atan(2))°
(1/sin(180-2atan(2))) = (c/sin(atan(2)))
c = 1.1180339887 units.
d² = 1.1180339887²+1²-2*1.1180339887cos(90+atan(2))
d = 2.0615528128 units.
e² = 2²+1²
e = √(5)
e = 2.2360679775 units.
f = 2.2360679775+1.1180339887
f = 3.3541019662 units.
g² = 3.3541019662²+2²-2*2*3.3541019662cos(atan(0.5))
g = 1.8027756377 units.
(1.8027756377/sin(atan(0.5))) = (3.3541019662/sinh)
h = 56.3099324748°
j = 180-56.3099324748
j = 123.6900675252°
k = 123.6900675252-atan(2)
k = 60.2551187023°
(2.0615528128/sin60.2551187023) = (1.8027756377/sinl)
l = 49.3987053536°
cosm = 1/2.0615528128
m = 60.9828593753°
n = 360-60.9828593753-49.3987053536-90-atan(0.5)
n = 133.0533840941°
tan(0.5*133.0533840941) = o/1
o = 2.3027756378 units.
Calculating Area Shaded.
It is;
2(0.5*2.3027756378-(0.5*133.0533840941π/360)
= 2(1.1513878189-0.5805549083)
= 1.1416658211 square units.
It implies;
Area Shaded ÷ Area Square to 2 decimal places is;
1.1416658211÷4
= 0.2854164553
≈ 0.29
Calculating Alpha.
Let it be p.
q² = 2-2cos133.0533840941
q = 1.8344914755 units.
r = 0.5(180-133.0533840941)
r = 23.473307953°
s = 90+23.473307953
s = 113.473307953°
t² = 2²+1.8344914755²-2*2*1.8344914755cos113.473307953
t = 3.2075272037 units.
(3.2075272037/sin113.473307953) = (1.8344914755/sinu)
u = 31.6415441908°
It implies;
Alpha, p to 1 decimal place is;
90-u-atan(1/2)
p = 90-31.6415441908-atan(0.5)
P = 31.7934046321°
≈ 31.8°
Therefore;
Shaded Area ÷ Area Square to 2 decimal places = 0.29
Angle Alpha to 1 decimal place = 31.8°
