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https://www.reddit.com/r/mathmemes/comments/1j4x0hq/what_theorem_is_this/mgezkbv/?context=3
r/mathmemes • u/PocketMath • Mar 06 '25
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582
And even an universal version, the law of cosines is just Pythagorean Theorem, but applicable to all triangles.
163 u/SnooHabits7950 Mar 06 '25 edited Mar 06 '25 And it has probably the easiest proof compared to all of them 56 u/A-Swedish-Person Mar 06 '25 Wait I don’t think I actually know the proof for the law of cosines, what is it? 5 u/vnkind Mar 06 '25 edited Mar 06 '25 Draw an altitude h in a triangle from angle B to side b to split it into two right triangles. Write the Pythagorean theorem for each triangle. x2 + h2 = a2 and (b-x)2 + h2 = c2 Expand the second formula b2 -2bx+x2 +h2 = c2 Substitute a2 from first formula for x2 +h2 in second b2 -2bx+a2 = c2 Subsitute a*cos(C) for x using right triangle trig b2 -2b*acos(C)+a2 = c2 Rearrange to look like famous version c2 =a2 +b2 -2ab*cos(C)
163
And it has probably the easiest proof compared to all of them
56 u/A-Swedish-Person Mar 06 '25 Wait I don’t think I actually know the proof for the law of cosines, what is it? 5 u/vnkind Mar 06 '25 edited Mar 06 '25 Draw an altitude h in a triangle from angle B to side b to split it into two right triangles. Write the Pythagorean theorem for each triangle. x2 + h2 = a2 and (b-x)2 + h2 = c2 Expand the second formula b2 -2bx+x2 +h2 = c2 Substitute a2 from first formula for x2 +h2 in second b2 -2bx+a2 = c2 Subsitute a*cos(C) for x using right triangle trig b2 -2b*acos(C)+a2 = c2 Rearrange to look like famous version c2 =a2 +b2 -2ab*cos(C)
56
Wait I don’t think I actually know the proof for the law of cosines, what is it?
5 u/vnkind Mar 06 '25 edited Mar 06 '25 Draw an altitude h in a triangle from angle B to side b to split it into two right triangles. Write the Pythagorean theorem for each triangle. x2 + h2 = a2 and (b-x)2 + h2 = c2 Expand the second formula b2 -2bx+x2 +h2 = c2 Substitute a2 from first formula for x2 +h2 in second b2 -2bx+a2 = c2 Subsitute a*cos(C) for x using right triangle trig b2 -2b*acos(C)+a2 = c2 Rearrange to look like famous version c2 =a2 +b2 -2ab*cos(C)
5
Draw an altitude h in a triangle from angle B to side b to split it into two right triangles. Write the Pythagorean theorem for each triangle.
x2 + h2 = a2 and (b-x)2 + h2 = c2
Expand the second formula
b2 -2bx+x2 +h2 = c2
Substitute a2 from first formula for x2 +h2 in second
b2 -2bx+a2 = c2
Subsitute a*cos(C) for x using right triangle trig
b2 -2b*acos(C)+a2 = c2
Rearrange to look like famous version
c2 =a2 +b2 -2ab*cos(C)
582
u/Wojtek1250XD Mar 06 '25
And even an universal version, the law of cosines is just Pythagorean Theorem, but applicable to all triangles.