The Law of sines
In any triangle, the sides are proportional to the sines of the opposite angles, i.e.,
a/sin A = b/sin B = c.sin C
Proof: Let ABC be any oblique triangle. Draw the altitude CD = h. Two cases are possible; all angles are acute or one angle obtuse.
In either case;
Sin B = h/a from the first figure
h = a sin B
From the second figure
sin A = h/b
h = b sin A
Equating the two values of h we get, asin B = bsin A => a/sin A = b/sin B
Similarly by drawing perpendicular from A to BC, we can prove that
c/sin C = b/sin B
Therefore in any triangle a/sin A = b/sin B = c/sin C
Relations between three sides and two angles
In any triangle, to prove that
a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A
In the first figure we have
AD/AC = cos A , hence AD = AC cos A = b cos A
DB/BC = cos B, hence DB = BC cos B = a cos B
Now, AB = AD + DB = b cos A + a cos B => c = a cos B + b cos A
Similarly, we can prove the other relations.
The law of cosines
The square on any side of a triangle is equal to the sum of the squares on the other two sides minus twice the product of those two sides and the cosine of the included angle, i.e,
a^{2} = b^{2} + c^{2}  2bc cos A
b^{2} = c^{2} + a^{2}  2ca cos B
c^{2} = a^{2} + b^{2}  2ab cos C
Proof: Consider any oblique triangle ABC with the altitude CD drawn from the vertex C to the opposite side.
From the first figure we have
AD = b cos A, CD = b sin A, DB = AB  AD = cb cos A
From figure 2 we have
cos A = cos (180  ang CAD) = cos CAD =  AD/b => AD = b cos A; CD = b sin A;
DB = DA + AB = c  b cos A
Thus in both figures, DB = c  b cos A, CD = b sin A
Hence a^{2} = DB^{2} + CD^{2} (Pythagoras theorem)
= (cb cos A)^{2} + b^{2} sin ^{2}A
=> a^{2} = c^{2}  2bc cos A + b^{2}(cos^{2}A + b^{2}Sin^{2}A).
Hence a^{2} = b^{2} + c^{2}  2bc cos A
Other results can be proved similarly.
Cos A = b^{2} + c^{2}  a^{2}/2bc
Cos B = c^{2} + a^{2}  b^{2}/2ca
Cos C = a^{2} + b^{2}  c^{2}/2ab
Area of a triangle
A = s(sa)(sb)(sc)^{1/2}
s = (a+b+c)/2
where a, b and c are sides of the triangle .
Directions: Solve the following.
