Mathematics Question and Solution
Let the required angle be x.
5 units, 3 units and 2 units are the radii respectively of each of the circles, decreasingly.
Calculating x.
a = 5+2
a = 7 units.
b = 5-2
b = 3 units.
c²+3² = 7²
c² = 49-9
c = √(40)
c = 2√(10) units.
cosd = 3/7
d = 64.6230664748°
e = 5+3
e = 8 units.
f = 3+2
f = 5 units.
5² = 8²+7²-2*7*8cosg
cosg = (8²+7²-5²)/(14*8)
g = 38.2132107017°
h = ½(180-(d+g))
h = ½(180-64.6230664748-38.2132107017)
h = 38.5818614118°
j = 90-h
j = 51.4181385882°
k² = 5²+5²-2*5*5cos(64.6230664748+38.2132107017)
k = 7.81717939862 units.
l² = 3²+3²-2*3*3cos(64.6230664748+38.2132107017)
l = 4.69030763917 units.
m = k+l
m = 12.5074870378 units.
sin51.4181385882 = n/12.5074870378
n = 9.77732700002 units.
cos51.4181385882 = o/12.5074870378
o = 7.80007107244 units.
p = o-c
p = 1.4755157521 units.
Therefore, the required angle x is;
x = asin(7.80007107244/12.5074870378)-atan(1.4755157521/9.77732700002)
x = 38.5818614118-8.58186141178
x = 30°
