The law of sines for an arbitrary triangle states:
If the sides of a triangle are a, b and c and the angles opposite those sides are A, B and C respectively, then
Sin A/a = Sin B/b = Sin C/c
or equivalently
a/Sin A = b/Sin B = c/Sin C = 2r
where r is the radius of the triangle’s circumcircle, which is the circle where all the vertices of the polygon manage to fit in.
Law of cosines
If angles A, B and C are opposite the sides a, b and c, then
c2 = a2 + b2 – 2ab cos(C)
b2 = a2 + c2 – 2ac cos(B)
a2 = b2 + c2 – 2bc cos(A)
This law of cosines generalizes the Pythagorean theorem, which holds only in right triangles- if the angle C is a right angle, its cosine is 0, and so the law of cosines reduces to
c2 = a2 + b2
Law of tangents
If the angles A, B and C are the angles opposite the sides a, b and c respectively, then
a – b/ a + b = tan[1/2(A- B)]/tan[1/2(A+B)]
Can you guys explain the proof of these laws?
2007-04-17
23:50:52
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2 answers
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asked by
Anonymous
in
Mathematics