Why are the following rules so? Trigonometry??

Question

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?

Answer 1

The law of sines can be proven by simply drawing a triangle, then using sine to write an expression for the height in terms of an angle and an adjacent leg. The first link gives a fairly simple proof.

I found an interesting proof for the law of cosines (2nd link).. see if you can fill in the blanks to complete it. There is a more direct proof of this by drawing an altitude in a triangle and writing expressions using the Pythagorean Theorem (3rd link).

You can use the law of sines to prove the law of tangents (4th link).

Answer 2

The derivation of the proofs require signs and symbols which cannot be drawn easily with standard input tools. Consult a good textbook on Trigonometry. S.L. Loney's books are the best.