First, you should know what the Law of Sines is. It is simply this:
" "a/sinA=b/sinB=c/sinC
Now let's look at our given. We have:
A=57°, a=11, and b=10
We need to look for B, C, and c.
Step 1 - Solving for B
Law of Sines
[1]" "a/sinA=b/sinB
Plug in the values of A, a, and b.
[2]" "11/sin(57°)=10/sinB
Multiply both sides by (sinB)[sin(57°)].
[3]" "11/cancelsin(57°)(sinB)cancel[sin(57°)]=10/cancelsinBcancel(sinB)[sin(57°)]
[4]" "11sinB=10sin(57°)
Divide both sides by 11.
[5]" "sinB=(10sin(57°))/11
Apply arcsin on both sides.
[6]" "arcsin(sinB)=arcsin((10sin(57°))/11)
[7]" "color(blue)(B=arcsin((10sin(57°))/11)~~49.678696410324°)
Step 2 - Solving for C
The sum of the interior angles of all triangles is 180°.
[1]" "A+B+C=180°
Isolate C.
[2]" "C=180°-A-B
Plug in the values of A and B.
[3]" "C=180°-57°-arcsin((10sin(57°))/11)
[4]" "color(blue)(C=123°-arcsin((10sin(57°))/11)~~73.321303589676°)
Step 3 - Solving for c
Law of Sines
[1]" "c/sinC=a/sinA
Multiply both sides by sinC
[2]" "c/cancelsinCcancel(sinC)=a/sinA(sinC)
[3]" "c=(asinC)/sinA
Plug in the values of A, a, and C.
[4]" "color(blue)(c=(11sin(123°-arcsin((10sin(57°))/11)))/sin(57°)~~12.564196743012485)