Mathematics Question and Solution
tana = 8/13
a = atan(8/13)°
tan(atan(8/13)) = b/10
b = 80/13 units.
It implies;
Area Blue1 is;
½*10(80/13)
= (400/13) square units.
= 30.7692307692 square units.
Calculating Area Blue2.
c² = 10²+8²
c = √(164)
c = 2√(41) units.
d² = 8²+13²
d = √(233) units.
233 = 9+164-12√(41)cose
12√(41)cose = 173-233
e = acos((173-233)/(12√(41)))
e = acos(-5/√(41))°
e = 141.3401917459°
f = e-90
f = (acos(-5/√(41))-90)°
tan(acos(-5/√(41))-90) = g/3
g = 3.75 units.
g = ¼(15) units.
h = 12-g
h = 12-¼(15)
h = ¼(48-15)
h = ¼(33) units.
tanj = ¼
j = atan(¼)°
tank = 3/(¼(15))
k = atan(⅘)°
l = j+k
l = (atan(¼)+atan(⅘))°
tanm = (¼(33))/3
m = atan(11/4)°
n = 90-j-m
n = (90-atan(¼)-atan(11/4))°
o² = 3²+(¼(33))²
o = ¼(3√(137)) units.
p = 180-(90-atan(¼)-atan(11/4))-(atan(¼)+atan(⅘))
p = (90+atan(11/4)-atan(4/5))°
p = 121.357085224°
(¼(3√(137))/sin(atan(¼)+atan(4/5))) = (q/sin(90-atan(¼)-atan(11/4)))
q = 1.1434150424 units.
It implies;
Area Blue2 is;
0.5*1.1434150424*¼(3√(137))sin121.357085224
= 4.2857142857
Therefore;
Total Blue Area exactly in decimal is ;
Area Blue1 + Area Blue2
= 30.7692307692+4.2857142857
= 35.0549450549 square units.
Calculating Cosec Alpha exactly.
Let Alpha be r.
Notice;
cosecr = 1/sinr
sinr = (33/4)/((3√(137)/4)
sinr = (33/(3√(137)))
It implies;
cosecr = (3√(137)/33)
cosecr = √(137)/11
Therefore, the answer;
Total Blue Area exactly in decimal is;
= 35.0549450549 square units.
Cosec Alpha exactly is;
= √(137)/11
