Find Angle A using Side a=3, Side b=4, Angle B=45 (Law of Sines)

Solution:
The law of sines states that
`a/sin(A)=b/sin(B)=c/sin(C)`

We have `a=3,b=4,B=45`

`:.a/sin(A)=b/sin(B)`

`:.sin(A)=(a*sin(B))/b`

`:.A=sin^(-1)((a*sin(B))/b)`

`:.A=sin^(-1)((3*sin(45))/4)`

`:.A=sin^(-1)((3*0.7071)/4)`

`:.A=sin^(-1)(0.5303)`

`:.A=32.0278`

`A+B+C=180`

`:.C=180-(A+B)`

`:.C=180-(32.0278+45)`

`:.C=180-77.0278`

`:.C=102.9722`

`c/sin(C)=b/sin(B)`

`:.c=(b*sin(C))/sin(B)`

`:.c=(4*sin(102.9722))/sin(45)`

`:.c=(4*0.9745)/(0.7071)`

`:.c=5.5125`

perimeter `P=a+b+c=3+4+5.5125=12.5125`

Area `=1/2 ab*sin(C)=1/2 *3*4*sin(102.9722)=1/2 *3*4*0.9745=5.8469`

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