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How do you prove using vector product cosA-B equals cos A cos B plus sin A sin B?
Wiki User
∙ 12y ago
First we need to use the laws of cosines to evaluate all of the angles of a certain triangle, then we need to go back and find what we evaulated for the cosine of A, the cosine of B, and we'll find the cosine of C just to make it easier. Basically this is saying that the cosine of A minus the cosine of B equals the cosine of A times the cosine of B PLUS the sine of A times the sine of B. All these letters are talking about sides, not angles. Let's say we had a triangle with the sides of 4, 5, and 7. Treat the angle opposite the side length of 4 Angle A, for the angle opposite the side length of 5 Angle B, and the last angle opposite the side length of 7 angle C.
a squared=b squared+c squared-2bc(cos)A
16=25+49-70cosA
16=74-70cosA
-70cosA=-58
cosA=-58/-70
cosA=0.82857142857143
Now let's use the inverse cosine to obtain the Angle opposite the side length of 4.
cos-1(0.82857142857143)=34.04773236999127832605
34.04773236999127832605 degrees is the measure of the angle opposite of the side length of 4.
b squared=a squared plus c squared-2ac(cos)b
25=16+49-56cosB
25=65-56cosB
-56cosB=-40
cosB=40/56
cosB=0.71428571428571
cos-1(0.71428571428571)=44.41530859719347303463
44.41530859719347303463 degrees is the measure of the angle opposite the side length of 5.
c squared=a squared plus b squared-2ab(cos)c
49=16+25-40cosC
49=41-40cosC
-40cosC=8
cosC=8/-40
cosC=-0.2
cos-1(-0.2)=101.53695903281392
101.53695903281392 degrees is the measure of the angle opposite the side length of 7.
Now that we have all of our angles in the triangle, we can use the laws of sines to check our work. Note that Sin A and Sin B means the sine of Angle A and the side of Angle B.
Sin A/ A=Sin B/B= Sin C/C (note that answers are approximate for sines)
34.04773236999127832605/4=8.5119330924978195815125
44.41530859719347303463/5=8.883061719438694606926
101.53695903281392 /7=14.50527986183056
Now let's just plug everything in into the original equation we have to check all of our work.
cos(0.82857142857143-0.71428571428571)=cos0.82857142857143*cos0.71428571428571+sin34.04773236999127832605*sin44.41530859719347303463
0.99989543723043-0.99992229248097=0.99989543723043*0.99992229248097+0.55988336977901*0.69985421222377
-0.00002685525054=-0.00002685525054
Congratulations. We have now proved our statement to be true.
Wiki User
∙ 12y ago
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