Basic Rules of Triangle
Semi-perimeter = a+b+c\over 2
(i) Sine Rule : sin\ A\over a=sin\ B\over b=sin\ C\over c=1\over 2R
where R is the radius of the circumcircle of ΔABC
(ii) Cosine Rule :
- cos A = b^2+c^2-a^2\over 2bc
- cos B = a^2+c^2-b^2\over 2ac
- cos C = a^2+b^2-c^2\over 2ab
(iii) Projection Rule
- a = b cos C + c cos B
- b = c cos A + a cos C
- c = a cos B + b cos A
(iv) Napier’s Analogy
- tanB-C\over 2=b-c\over b+ccot A\over 2
- tanC-A\over 2=c-a\over c+acot B\over 2
- tanA-B\over 2=a-b\over a+bcot C\over 2
Half Angles of T. Ratio
| sinA\over 2= \sqrt{(s-b)(s-c)\over bc} | cosA\over 2= \sqrt{s(s-a)\over bc} |
| sinB\over 2= \sqrt{(s-c)(s-a)\over ac} | cosB\over 2= \sqrt{s(s-b)\over ac} |
| sinA\over 2= \sqrt{(s-a)(s-b)\over ab} | cosC\over 2= \sqrt{s(s-c)\over ab} |
Area of a Triangle
Consider a triangle of sides a, b and d.
(i) Δ = ½ bc sin A = ½ ca sin B = ½ ab sin C
(ii) Δ = c^2 sin A sin B\over 2 sin C
(iii) Δ = a^2 sin B sin C\over 2 sin A
(iv) Δ = b^2 sin C sin A\over 2 sin B
(v) Δ = \sqrt{s(s-a)(s-b)(s-c)}
where, a+b+c\over 2
