4. For this triangle, the sides a = 1.5, b = 1.7, and angle B = 46°. Use the law of Sines to find angle A.
a = b
Sin A Sin B
1.5 = 1.7
Sin A Sin46°
1.7SinA = 1.5Sin46°
SinA = 1.5Sin46°
1.7
Sin A = 0.6347
A = Sin-1 0.6347
ÐA = 39°
5. You can also apply the law of sines and cosines to other shapes when trying to find area, in such things as the example shown below. First you must separate the shape into two triangles. Then you should find Angle C in the other triangle because it it needed in the formula for area, which is Area = ( ½ (ab) SinC)
First using the Law of Cosines, we can find the missing side.
c2 = a2 + b2 – 2ab CosC
c2 = 47 + 52 – 2(47)(52) Cos98°
c2 = 2209 + 2704 – 2(2444) Cos98°
c2 = 4913 – 4888 Cos98°
c2 = 4913 + 680.28
c2 = 5593.28
c = 74.8
Now that you know the measure of side c, you can again use the Law of Cosines to find the opposite angle being Angle C.
c2 = a2 + b2 – 2ab CosC
74.82 = 392 + 722 – 2(39)(72) CosC
5595 = 1521 + 5184 – 2(2808) CosC
5595 = 6705 – 5616 CosC
-1110 = -5616 CosC
0.1976 = CosC
Cos-1 0.1976 = C
ÐC = 79°
Another formula that can be used to find an angle using the Law of Cosines is CosC = a2 + b2 – c2
2 (a)(b)
Now using the area formula we can figure out the area of the two triangles and combine the areas.
Area(1) = ½ (ab) SinC
Area = ½ (47)(52) Sin98°
Area = ½ 2444 Sin98°
Area = ½ 2420
Area(1) = 12102
Area(2) = ½ (ab) SinC
Area = ½ (39)(72) Sin 79°
Area = ½ (2808) Sin 79°
Area = ½ 2756
Area(2) = 13782
Now by combining area(1) and area(2) you get the total area.
Area(1) + Area(2) = Area(T)
12102 + 13782 = 25882
The total area equals 2588 units squared.
6. Other examples to get you familiar with using both the Law of Sines as well as the Law of Cosines are in such triangles where the three sides are known, in this case side a = 5, side b = 6, and side c = 8, you should find all angles. Knowing all sides means that you must use the Law of Cosines to start.
a2 = b2 + c2 – 2 bc CosA
52 = 62 + 82 – 2 (8)(6) CosA
25 = 36 + 64 – 2 (48) Cos A
25 = 100 – 96 Cos A
-75 = -96 CosA
0.78125 = CosA
Cos-1 0.78125 = A
ÐA = 39°
You now know one of the angles of the triangle, so you can apply the Law of Sines to find the other missing angles.
a = b
SinA SinB
5 = 6
Sin39° SinB
5SinB = 6Sin39°
SinB = 6Sin39°
5
SinB = 0.7552
B =Sin-1 0.7552
ÐB = 49°
And since you now know two of three possible angles, you know that a triangle is made up to 180°, therefore you can subtract the two known angles from 180 to get the final missing angle.
180 – 49 – 39 = 92
ÐC = 92°
