The solution of triangle formula sheet is a set of formulas and equations used to solve various problems related to triangles. These formulas help in finding the measurements of angles, sides, area, perimeter, and other important properties of a triangle. Some of the common formulas include the Pyt...
SOLUTIONS OF TRIANGLE
I. SINE FORMULA : In any triangle ABC , a b c .
sin A sin B sin C
b 2 c 2 a 2
II. COSINE FORMULA : (i) cos A = or a² = b² + c² 2bc. cos A
2bc
c 2 a 2 b 2 a 2 b 2 c 2
(ii) cos B = (iii) cos C =
2ca 2ab
III. PROJECTION FORMULA : (i) a = b cos C + c cos B (ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A
BC bc A
IV. NAPIER’S ANALOGY TANGENT RULE : (i) tan = cot
2 b c 2
CA ca B AB a b C
(ii) tan = cot (iii) tan = cot
2 c a 2 2 a b 2
V. TRIGONOMETRIC FUNCTIONS OF HALF ANGLES :
A (sb)(sc) B (sc)(sa ) C (sa )(sb)
(i) sin = ; sin = ; sin =
2 bc 2 ca 2 ab
A s(sa ) B s(sb) C s(sc)
(ii) cos = ; cos = ca ; cos =
2 bc 2 2 ab
A (sb)(sc) a b c
(iii) tan = = where s = & = area of triangle.
2 s(sa ) s(sa ) 2
(iv) Area of triangle = s(sa )(sb)(sc) .
VI. MN RULE : In any triangle ,
(m + n) cot m cot n cot
n cot B m cot C
1 1 1
VII. ab sin C = bc sin A = ca sin B = area of triangle ABC.
2 2 2
a b c
= 2R
sin A sin B sin C
a bc
Note that R = 4 ; Where R is the radius of circumcircle & is area of triangle
VIII. Radius of the incircle ‘r’ is given by:
a bc A B C
(a) r = where s = (b) r = (s a) tan = (s b) tan = (s c) tan
s 2 2 2 2
a sin B2 sin C2 A B C
(c) r = & so on (d) r = 4R sin sin sin
cos A2 2 2 2
IX. Radius of the Ex circles r1 , r2 & r3 are given by :
A B C
(a) r1 = ; r2 = ; r3 = (b) r1 = s tan ; r2 = s tan ; r3 = s tan
sa sb sc 2 2 2
a cos B2 cos C2 A B C
(c) r1 = & so on (d) r1 = 4 R sin . cos . cos ;
cos A2 2 2 2
B A C C A B
r2 = 4 R sin . cos . cos ; r3 = 4 R sin . cos . cos
2 2 2 2 2 2
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