Basics of Solutions of Triangles Flashcards
Sine Law
a/sinA = b/sinB = c/sinC= 2R, where R= circumradius of the triangle, a, b, c are the length of the triangle and A, B, C are the angle between the sides
Cosine Law
cosA = b^2+c^2-a^2/2bc and vice versa
Projection Formula
a = bcosC+ccosB
and vice versa
This can be proved by taking components in a triangle
Tangent Law(Napier’s Analogy)
tan(A-B/2) = (a-b/a+b)cotC/2 and vice versa
Formula of semi perimeter
s = (a+b+c)/2, where a, b, c are the length of the triangle
Find area using heron’s formula
root[s(s-a)(s-b)(s-c)]
sinA/2
root[(s-b)(s-c)/bc]
cosA/2
root[s(s-a)/bc]
tanA/2
root[(s-b)(s-c)/s(s-a)]
Area of triangle based on product of sides and included angle
Area = Product of two sides * included angle /2
Circumcircle and Circumcentre
The circle passing through all the vertices of the triangles is called as circumcircle and the intersection of perpendicular bisectors of the sides are circumcentre.
Relation between circumradius, lengths of sides and area of the triangles
R =abc/4Δ
Relation between lengths of sides and the circumradius for an equilateral triangle
a^2+b^2+c^2 = 9R^2
Relation between lengths of sides and the circumradius for an right-angled triangle
a^2+b^2+c^2 = 8R^2
Incircle and incentre
The circle touches all three sides of the triangle internally and the centre is the point of intersection of internal angle bisector.
