Using the "Law of Sines" to solve the triangle.?

Question

A=44°
B=25°
a=12

Answer 1

The law of sines is:
a/sinA = b/sinB = c/sinC

You are given both a and sinA, so figure out that first:
a/sinA = 12/sin(44) = 17.274678...

You know that b/sinB is the same, and since you already know B, you can solve for b easily:

b/sinB = 17.274678...
b/sin(25) = 17.274678...
b = 7.3005946...

And you know C since the sum of all three angles in a triangle is 180, and you know the other two angles, and can therefore solve for c:

c/sinC = 17.274678...
c/sin(180 - 44 - 25) = 17.274678...
c/sin(111) = 17.274678...
c = 16.1270317...

Answer 2

You can easily solve for angle C (or γ): γ = 180° - 25° - 44° = 111°

The Law of Sines states that: a / sin(α) = b / sin(β) = c / sin(γ) - So just plug in your values and solve for b

a / sin(α) = b / sin(β)

b = 12 * sin(β) / sin(α)

b = 12 * sin(25°) / sin(44°)

b = 7.3 (approx)

Now that you have b, use the Law of Sines again to solve for side c.

a / sin(α) = c / sin(γ)

c = a * sin(γ) / sin(α)

c = 12 * sin(111°) / sin(44°)

c = 16.127 (approx)

Hope that helps.