use the information to solve the triangle. If two solutions exist, find both solutions. How do you work this out?

1.A=36°, B=98°, c=16
2.a=4,b=8,c=10
3.A=35°,b=8,c=12
4.A=25°,b=28,a=18
5.B=130°,c=10.1,b=5.2
6.A=150°,b=4.8,a=9.4

2 answers

if given two angles, use A+B+C=180°
Then, if only one side, use law of sines

If three sides, use law of cosines to find an angle.

If only one angle, use law of sines to get a 2nd one. Then see step 1.
I assume you are familiar with the tradional way of identifying the information given about a triangle, such as ASA, SAS, SSS and SSA
use "cosine law" for SAS and SSS
use "sine law" for ASA and SSA

#1 - a case of ASA --> a side contained between two angles. We can find the third angle and then use the sine law
Angle C = 46°
a/sin36 = 16/sin46
a = 16sin36/sin45 = appr 13.1
find b in the same way

#2, you have all three sides, so SSS and cosine law.
I always find the smallest angle (opposite the smallest side). That way I don't get confused about the cosine of an obtuse angle being negative
4^2 = 8^2 + 10^2 - 2(8)(10)cosA
160cosA = 64+100-16
cosA = 148/160 = 37/40
angle A = appr 22.33°
Now use the sine law to find a second angle, then use the supplementary angle propery to find the third.

#3 -- cosine law to find a, then use the sine law to find a second angle

#4. a case of SSA, which could be trouble.
let's find angle B
sinB/28 = sin25/18
sinB = .6574..
B = 41.1° or B = 180-41.1 = 138.9°

Case1: Angle B=41.1, Angle A = 25°, so angleC= 113.9°
c/sin113.9 = 18/sin25
c = 38.9

case2: angle B = 138.9, A=25°, C = 16.1°
c/sin16.1 = 18/sin25
c = 11.8

#5 another SSA --- try the sine law
sinC/10.1 = sin130/9.4
sinC = .823..
C = 55.4° or 124.6°

case1: C = 55.4, B=130, which already is more than 180°, so this is not possible

case2: C=124.6, B = 130, way over 180 already.

So this triangle is not possible. (just because we can write down a bunch of data does not mean we can actually draw the triangle.
Try drawing a triangle with the original given data. )

#6
again make your sketch. case of SSA

sinB/4.8 = sin150/9.4
sinB = .253..
B = 14.8° or 165.2°

case1: A = 150, B = 14.8, then C = 15.2°
c/sin15.2 = 9.4/sin150
c = 4.9

case2: A = 150, B = 165.2 ---> not possible, over 180°

notice the problems arise when we have SSA
solving for the angle results in two cases, since the sine is positive in both the first and second quadrants.