Use the Law of Cosines to solve the triangle Round your answ

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.

B = 120° 20\',  a = 34,  c = 34

A=

b=

C=

Solution

We know that law of cosine is

b^2= a^2 + c^2 -2ac Cos B

Now we substitute the given values of B,a and c and solve for b.

b^2=34^2 + 34^2 -2*34*34 cos 120

b^2= 1156 + 1156 -2312 cos 120

b^2= 2312 - 2312 cos120

b^2= 2312 - 2312(-0.5)

b^2=2312+1156

b^2=3468

To solve for b,we have to take square root both sides

b=+- square root 3468

b=+- 58.88972 approximately

b=+- 58.89

We have to ignore negative value since side cant be negative

b=58.89

Now we use the same cosine law and solve for A

a^2= b^2 + c^2 -2bc cos A

Next step is to substitute the values of a, b and c in this formula

34^2= 58.89^2 + 34^2 -2*58.89*34 cos A

1156= 3468 + 1156 - 4004.52 cos A

1156= 4624 -4004.52 cos A

Next step is to move 4624 to the left side of the equation and for that we have to subtract 4624 to both sides

1156-4624= 4624-4624 -4004.52 cos A

-3468= -4004.52 cos A

Next step is to divide both sides by -4004.52

-3468/-4004.52=-4004.52 cos A/-4004.52

.866= cos A

And to solve for x,we take cos ^-1 on both sides

cos ^-1.866= cos ^-1 (cos A)

30=A

Therefore A= 30 degree

We have B=120 and we get A=30

And sum of angles of a triangle is 180

Therefore A+B+C=180

We substitute the values of A and B and solve for C

30 + 120+C=180

150 + C=180

To solve for C, we have to subtract 150 to both sides

150-150+C=180-150

C=30 degree

Hence our answer is

A=30 degree

b=58.89

C=30 degree