Examples:

example # 1 Law of Cosines: In order for Mary to travel to work from home she has to get on her bike and go 20 km directly east, and then she has to go 10 km in a direction that is directly north-east (45º north of east). The county that she lives in is trying to decide whether they should build a road that goes straight from the area where she lives to the area where she works at. How many kilometers of travel will Mary save on each trip if this road is built?

Solution: If we draw a schematic figure for this situation we note that immediately we are encoutnered with a the situation in which we have not so "neat" a triangle! Part 1 of the path to work, going directly east, is labeled as a = 20 km. Part 2 of the path, going NE, is labeled as b = 10 km. The fact that Part 2 points 45º north of east allows us to calculate the angle q inside the triangle, as shown below. We simply subracact 45 from 180.

Now we know a, we know b, and we know the angle between them. We can put this information in the law of cosines to get the side c of the triangle:

c2 = a2 + b2 - 2a.b.Cos(q) c2 = (20 km)2 + (10 km)2 - 2.(10km)(20km)Cos (135º) c2 = 400 km2 + 100 km2 - 400 x (-0.7071) km2 c2 = 500 km2 + 282.84 km2 c2 = 782.84 km2

Taking the square root of both sides we get: c = 28 km. Note that I have rounded off the final answer to two significant figures and that the units came out in km, which is expected. Thus, the person will save about 2 kilometers in each direction since without the alternate path she will be traveling 30 km.

example#2 Law of Sines: Compute the side opposite the given angle, compute the third angle by subtracting the given angles from 180 degrees, then compute the side opposite that angle.

Trig is fun!

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