Answer:
The Law of Sines applies to any triangle and works as follows:
a/sinA = b/sinB = c/sinC
We are attempting to solve for every angle and every side of the triangle. With the given information, A = 61°, a = 17, b = 19, we can solve for the unknown angle that is B.
a/sinA = b/sinB
17/sin61 = 19/sinB
sinB = (19/17)(sin61)
sinB = 0.9774
sin-1(sinB) = sin-1(0.9774)
B = 77.8°
With angle B we can solve for angle C and then side c.
A + B + C = 180°
C = 180° - A - B
C = 180° - 61° - 77.8°
C = 41.2°
a/sinA = c/sinC
17/sin61 = c/sin41.2
c = 17(sin41.2/sin61)
c = 12.8
The first solved triangle is:
A = 61°, a = 17, B = 77.8°, b = 19, C = 41.2°, c = 12.8
However, when we solved for angle B initially, that was not the only possible answer because of the fact that sinB = sin(180-B).
The other angle is simply 180°-77.8° = 102.2°. Therefore, angle B can also be 102.2° which will give us different values for c and C.
C = 180° - A - B
C = 180° - 61° - 102.2°
C = 16.8°
a/sinA = c/sinC
17/sin61 = c/sin16.8
c = 17(sin16.8/sin61)
c = 5.6
The complete second triangle has the following dimensions:
A = 61°, a = 17, B = 102.2°, b = 19, C = 16.8°, c = 5.6
The answer you are looking for is the first option given in the question:
B = 77.8°, C = 41.2°, c = 12.8; B = 102.2°, C = 16.8°, c = 5.6
Explanation: