Find the possible values of the included angle of a triangle with: a) sides of length 5 cm and 8 cm, and area of 15 cm^2 b) sides of length 45 km and 53 km, and area 800 km^2.

We know that

The area of a triangle can be calculated by applying the law of sines

Area=(1/2)*a*b*sin C> sin C=2*A/[a*b]

Part a) sides of length 5 cm and 8 cm, and area of 15 cm^2
a=5 cm
b=8 cm
A=15  cm²
sin C=2*A/[a*b]> sin C=2*15/[5*8]> sin C=3/4
C=arc sin (3/4)> C=48.59 degrees
the possible values of the included angle are
C1=48.59°
C2=180°-48.59> C2=131.41°

the answer Part a) is
48.59° and 131.41°

Part b) sides of length 45 km and 53 km, and area 800 km^2
a=45 km
b=53 km
A=800 km²
sin C=2*A/[a*b]> sin C=2*800/[45*53]> sin C=0.6709
C=arc sin (0.6709)> C=42.13°
C1=42.13°
C2=180-42.13°> C2=137.87°

the answer part b) is
42.13° and 137.87°