Solve the triangle. A = 32°, a = 19, b = 12?

a/sinA = b/sinB (sine law)
19/sin32° = 12/sinB
sinB = 12 sin32° / 19
B = 19.6°

A + B + C = 180° (sum of interior angles of triangle)
32° + 19.6° + C = 180°
c = 28.1


The answer: B. B = 19.6°, C = 128.4°, c ≈ 28.1

sin(32)/19 = sin(B)/12
cross multiply
12 sin(32) = 19 sin(B)
sin(B) = 12 sin(32) /19 = 0.334686

B = sin°-1(0.334686)
B = 0.321272 radian
B = (0.321272)*180/pi degrees = 19.55°

C = 180-32-19.55 = 128.45°

sin(32)/19 = sin(128.45)/ c
cross multiply
c sin(32) = 19 sin(128.45)
c = 19 sin(128.45) /sin(32) =28.08

https://gyazo.com/a9667c8a97669f974ac9be2f95b87011

Use The Law of Sines:

[a / sin(A)] = [b / sin(B)] # Or you can flip them: [sin(A) / a] = [sin(B) / b]
Note: Capital "A" is a certain angle of a triangle, and lowercase "a" is its side,
and Capital "B" is another certain angle of a triangle, and lowercase"b" is its opposite side.
So then, now that we know the principle, not worrying about "A", "B", "C", "a", "b", "c", let's just be consistent.

[sin(B) / b] = [sin(A) / a]
We want angle B, so isolate sin(B), by multiplying out b to both sides of the equation, and we get this:

sin(B) = (b / a) * sin(A)
we need the angle, so to get that, we cancel the sine function with the arcsine function:

B = arcsin[(b / a) * sin(A)]

Make sure your calculator is in degree mode, not radian mode.
= arcsin[(12 / 19) * sin(32 degrees)]

= arcsin(0.334685851)

Angle B =~ 19.5534 degrees

There are 180 degrees in a triangle so: angle A + angle B + angle C = 180 degrees,
so then: angle C = 180 - angle A - angle B

C = 180 degrees - 32 degrees - 19.5534 degrees

C = 128.4466 degrees

We also need to know c, and we should not make any assumptions such as that this is a right triangle and so we can use the Pythagorean Theorem. It's not from what we know that angle C is greater than 90 degrees. So using the law of sines again:

[c / sin(C)] = [a / sin(A)] # or [b / sin(B)] is also valid.
So then:

c = [a / sin(A)] * sin(C)

= [19 / sin(32°)] * sin(128.4466)

=~ 28.1°

*************
So then we know: A = 32°, a = 19, B = 19.6°, b = 12, C = 128.4°, c = 28.1°

The only answer that satisfies all these determined values is that the answer is ...

Solution: B)

19/sin 32 = 12/sinB
sin B = 12 sin 32/19
B = 19.6°

C = 180° - 32° - 19.6°
C = 128.4°

c/sin 128.4° = 19/sin 32°
c = 19 sin 128.4°/ sin 32°
c = 28.1

B = 180° - 128.4° - 32°
B = 19.6°