1. Side a = 3, side b = 5, and side c = 7. Find all missing angles. In this case, 3 sides are known so you should use the law of cosines.
c2 = a2 + b2 - 2ab Cos C
49 = 9 + 25 - 2(3)(5) Cos C
49 = 34 - 30 Cos C
15 = -30 Cos C
-1/2 = Cos C
Cos-1 (-1/2) = C
120 = C
ÐC = 120°
b2 = a2 + c2 – 2ac CosB
25 = 9 + 49 – 2(21) CosB
25 = 58 – (42) CosB
-33 = (-42) CosB
.7857 = CosB
Cos-1 (.7857) = B
38 = B
ÐB = 38°
A2 = b2 + c2 – 2bc CosA
9 = 25 + 49 – 2(5)(7) CosA
9 = 74 – 2(35) CosA
-65 = (-70) CosA
.92857 = CosA
Cos-1 (.92857) = A
22 = A
ÐA = 22°
2. Side a = 4, side b = 6, and angle A = 40°. Find angle B. In this case, 2 sides and an angle opposite one of the sides are known, so you can use the law of sines.
a = b
Sin A Sin B
4 = 6
Sin 40° Sin B
4 (Sin B) = 6 (Sin 40°)
Sin B = 6 (Sin 40°)
4
Sin B = 0.964
ÐB = 75°
3. If side a = 8, side b = 11, and side c = 14, what are the missing angles? You again have three sides, so the law of cosines should be used.
b2 = a2 + c2 - 2ac CosB
121 = 64 + 196 – 2(112) CosB
121 = 260 – (224) CosB
-139 = -224 CosB
.6205 = CosB
Cos-1 (.6205) = B
52° = B
ÐB = 52°
a2 = b2 + c2 – 2bc CosA
64 = 121 + 196 – 2(154) CosA
64 = 317 – (308) CosA
-253 = (-308) CosA
.8214 = CosA
Cos-1 (.8214) = A
34.77 = A
ÐA = 35°
Now angle C can be found from subtracting 35° and 52° from 180°, this equals
180° - 35° – 52° = 93°
(However you can check that this is right by again using the law of cosines).
c2 = a2 + b2 – 2ab CosC
196 = 64 + 121 – 2(88) CosC
196 = 185 – (176) CosC
11 = (- 176) CosC
-.0625 = CosC
Cos-1 (-.0625) = C
93 = C
ÐC = 93°
