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waterdemon

Properties and solution of triangles :

r = 4R SinA/2.SinB/2.SinC/2

Sine rule :

a/Sina = b/Sinb = c/Sinc = 2R

Cosine rule :

CosA = b²+c²-a²/2bc

Projection rule :

a = bCosC + cCosB

Napier's rule :

tan(A-B/2) = [(a-b)/(a+b)]Cot C/2

Half angle Formulae :

Sin A/2 = [(s-b)(s-c)/bc]^1/2

Cos A/2 = [s(s-a)/bc]^1/2

tan A/2 = [(s-b)(s-c)/s(s-a)]^1/2

Where 2s = a+b+c

Circumradius :

R = abc/4

Inradius :

r = /s = (s-a)tan A/2

Exradius :

r₀ = /(s-a) = s tan A/2

Solution of triangles :

1)given a,b,c apply Cosine rule or half angle rule.

2)given b,c,A apply Napier's rule.

3)given a,B,C apply sine rule.

4)given b,c,B apply sine rule to obtain sinC (cases):

For acute B:

if SinC > 1 No triangle

if SinC = 1 one right angled triangle.

if SinC < 1 & cSinB < b < c Two triangles.

if SinC < 1 & b c one triangle.

For Obtuse :

if b > c one triangle.

if b c no triangle.

Area of triangle :

= 1/2(bcSinA) = [s(s-a)(s-b)(s-c)]^1/2

Area of cyclic quadrilateral ABCD =

[(s-a)(s-b)(s-c)(s-d)]^1/2

Where 2s = a+b+c+d.

Ptolmey's theorem :

AC.BD = AB.CD + BC.AD

Area of n sided regular polygon

= nR²/2[Sin(2/n)]

Where R = radius of circumsircle.

= nR²[tan(/n)]

Hope you all find it useful.

Cheers !!!!!!!!!!!!!!!!!

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