**Solution**

**1) B=32.7°, a=37.5cm, b=28.6cm**

**this is a special case so, first**

**use law of sine to find sinA**

**sinA/a=sinB/b**

**sinA=a×sinB/b=37.5×sin32.7°/28.6=0.7384**

**sinA<1 therefore, two triangles are possible**

**And A=arcsin(0.7384)=47.6°**

**and also, sin is positive in second quadrant**

**so, A=180°-47.6°=132.4°**

**now, solve first triangle with A=47.6°**

**By angle sum property of triangle**

**C=180°-A-B=180°-47.6°-32.7°=99.7°**

**use law of sine**

**sinC/c=sinB/b**

**c=b×sinC/sinB=28.6×sin99.7°/sin32.7°=52.2**

**now, solve second triangle with A=132.4°**

**by angle sum property of triangle**

**C=180°-132.4°-32.7°=14.9°**

**use law of sine**

**c=b×sinC/sinB=28.6×sin14.9°/sin32.7°=13.6**

**so, other values of triangle 1 are**

**A=47.6°, C=99.7° and c=52.2cm**

**values of second triangle are**

**A=132.4°, C=14.9° and c=13.6cm**

**to find the area use heron's formula**

**s=(a+b+c)/2 , A=√[s(s-a)(s-b)(s-c)]**

**here, a=37.5, b=28.6 and c=52.2 (for first triangle)**

**s=(37.5+28.6+52.2)/2=59.2**

**s-a=59.2-37.5=21.7**

**s-b=59.2-28.6=30.6**

**s-c=59.2-52.2=7**

**so, area=A=√[59.2×21.7×30.6×7]=√275169.88=524.6 (cm)^2**

**for second triangle**

**a=37.5 , b=28.6 and c=13.6**

**s=(37.5+28.6+13.6)/2=39.9**

**s-a=39.9-37.5=2.4**

**s-b=39.9-28.6=11.3**

**s-c=39.9-13.6=26.3**

**so, area=A=√[39.6×2.4×11.3×26.3]=√28423.3=168.6 cm^2**

**b)a=4.1in, b=9.8in, c=6.2in**

**first of all area by heron's formula**

**s=(4.1+9.8+6.2)/2=10.1**

**s-a=10.1-4.1=6**

**s-b=10.1-9.8=0.3**

**s-c=10.1-6.2=3.9**

**A=√[10.1×6×0.3×3.9]=√70.551=8.4 in^2**

**now, by law of cosine**

**a^2=b^2+c^2-2bc×cosA**

**cos A=-(a^2-b^2-c^2)/2bc**

**cosA=-[(4.1)^2-(9.8)^2-(6.2)^2]/2×9.8×6.2**

**cosA=117.7/121.52=0.9685**

**A=arccos(0.9685)=14.4°**

**now, usw law of sine**

**sinA/a=sinB/b**

**sinB=b×sinA/a=9.8×sin14.4°/4.1=0.5944**

**B=arcsin(0.5944)=36.5°**

**use angle sum property of triangle**

**C=180°-14.4°-36.5°=129.1°**

**so, other values are**

**A=14.4°, B=36.5°, C=129.1° and Area=8.4 in^2**