Law of Sines calculator

SolutionHelp

1. Find Side a using Side b = 3, Angle A = 60, Angle B = 45

2. Find Side b using Side a = 3, Angle A = 60, Angle B = 45

3. Find Angle A using Side a = 3, Side b = 4, Angle B = 45

4. Find Angle B using Side a = 3, Side b = 4, Angle A = 45

Example

1. Find Side a using Side b=3, Angle A=60, Angle B=45 (Law of Sines)

Solution:
The law of sines states that

asin(A)=bsin(B)=csin(C) $a/sin(A)=b/sin(B)=c/sin(C)$

We have

b=3,A=60,B=45 $b=3,A=60,B=45$

∴asin(A)=bsin(B) $:.a/sin(A)=b/sin(B)$

∴a=b⋅sin(A)sin(B) $:.a=(b*sin(A))/sin(B)$

∴a=3⋅sin(60)sin(45) $:.a=(3*sin(60))/sin(45)$

∴a=3⋅0.8660.7071 $:.a=(3*0.866)/(0.7071)$

∴a=3.6742 $:.a=3.6742$

A+B+C=180 $A+B+C=180$

∴C=180-(A+B) $:.C=180-(A+B)$

∴C=180-(60+45) $:.C=180-(60+45)$

∴C=180-105 $:.C=180-105$

∴C=75 $:.C=75$

csin(C)=bsin(B) $c/sin(C)=b/sin(B)$

∴c=b⋅sin(C)sin(B) $:.c=(b*sin(C))/sin(B)$

∴c=3⋅sin(75)sin(45) $:.c=(3*sin(75))/sin(45)$

∴c=3⋅0.96590.7071 $:.c=(3*0.9659)/(0.7071)$

∴c=4.0981 $:.c=4.0981$

perimeter

P=a+b+c=3.6742+3+4.0981=10.7723 $P=a+b+c=3.6742+3+4.0981=10.7723$

Area

=12ab⋅sin(C)=12⋅3.6742⋅3⋅sin(75)=12⋅3.6742⋅3⋅0.9659=5.3236 $=1/2 ab*sin(C)=1/2 *3.6742*3*sin(75)=1/2 *3.6742*3*0.9659=5.3236$