Question

2. Given A=15°, b=8, and a=6, use the Law of Sines to solve the triangle (if possible)

for the value of c. If two solutions exist, find both. Round answer to two decimal places.
A) c=11.01
B) c=2.10 and 13.36
C) c=12.14
D= c=1.12 and 10.24
E) not possible

Answer 1

B

Explanation:

The Law of Sines states that

sinA/a = sinB/b = sinC/c

We know A, a, and b. Given this, we can solve for B, and given that, we can solve for C and then c.

First, we solve for B.

sinA/a = sinB/b

sin(15°)/6 = sinB/8

sin(15°)/6 * 8 = sinB

arcsin(sin(15°)/6 * 8) = B ≈ 20.187

Note that there can be a second B value. We can check if there is a second B value by first subtracting the B value we got from 180.

180 - 20.187 = 159.813

Next, we can add that to our existing angle, A = 15°. If that is less than 180, then we do have a second solution. 159.813 + 15 = 174.813 < 180 so there does exist a second solution where B = 159.813

Next, we can solve for C.

We know that the angles in a triangle add to 180°, so A + B + C = 180. Therefore, for the first solution:

15 + 20.187 + C = 180

C = 180 - 15 - 20.187 = 144.813°

and for the second:

15 + 159.813 + C = 180

C = 180 - 15 - 159.813 = 5.187

Finally,

sinA/a = sinC/c

multiply both sides by c to no longer have it as a denominator

sinA * c / a = sinC

divide both sides by (sinA/a) to isolate c

for the first solution:

c = sinC*a/sinA = 13.35868 ≈ 13.36

and for the second:

c = sinC*a/sinA ≈ 2.10