Important Formulas: Solutions of Triangles

 Page 1


Page # 41
SOLUTION OF TRIANGLE
1. Sine Rule:
C sin
c
B sin
b
A sin
a
? ? .
2. Cosine Formula:
(i) cos
 
A =
b c a
b c
2 2 2
2
? ?
   (ii)  cos
 
B = 
c a b
ca
222
2
? ?
(iii)  cos
 
C = 
a b c
a b
2 2 2
2
? ?
3. Projection  Formula:
(i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA
4. Napier’s Analogy - tangent rule:
(i) tan 
2
C B ?
 = 
c b
c b
?
?
 cot 
2
A
(ii)  tan
2
A C ?
 =
a c
a c
?
?
 cot
B
2
(iii)  tan
A B ?
2
 =
a b
a b
?
?
 cot
C
2
5. Trigonometric Functions of Half Angles:
(i) sin
A
2
 =
( ) ( ) s b s c
b c
? ?
 ; sin
B
2
 =
( ) ( ) s c s a
ca
? ?
 ;
sin
C
2
 = 
( ) ( ) s a s b
a b
? ?
(ii) cos
A
2
 =
s s a
b c
( ) ?
 ; cos
B
2
 =
s s b
ca
( ) ?
 ;  cos
C
2
 = 
s s c
a b
( ) ?
(iii) tan
A
2
 = 
( ) ( )
( )
s b s c
s s a
? ?
?
=
?
s s a ( ) ?
 where s =
a b c ? ?
2
 is semi
perimetre of triangle.
(iv) sin A = ) c s)( b s)( a s ( s
bc
2
? ? ? = 
bc
2 ?
Page 2


Page # 41
SOLUTION OF TRIANGLE
1. Sine Rule:
C sin
c
B sin
b
A sin
a
? ? .
2. Cosine Formula:
(i) cos
 
A =
b c a
b c
2 2 2
2
? ?
   (ii)  cos
 
B = 
c a b
ca
222
2
? ?
(iii)  cos
 
C = 
a b c
a b
2 2 2
2
? ?
3. Projection  Formula:
(i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA
4. Napier’s Analogy - tangent rule:
(i) tan 
2
C B ?
 = 
c b
c b
?
?
 cot 
2
A
(ii)  tan
2
A C ?
 =
a c
a c
?
?
 cot
B
2
(iii)  tan
A B ?
2
 =
a b
a b
?
?
 cot
C
2
5. Trigonometric Functions of Half Angles:
(i) sin
A
2
 =
( ) ( ) s b s c
b c
? ?
 ; sin
B
2
 =
( ) ( ) s c s a
ca
? ?
 ;
sin
C
2
 = 
( ) ( ) s a s b
a b
? ?
(ii) cos
A
2
 =
s s a
b c
( ) ?
 ; cos
B
2
 =
s s b
ca
( ) ?
 ;  cos
C
2
 = 
s s c
a b
( ) ?
(iii) tan
A
2
 = 
( ) ( )
( )
s b s c
s s a
? ?
?
=
?
s s a ( ) ?
 where s =
a b c ? ?
2
 is semi
perimetre of triangle.
(iv) sin A = ) c s)( b s)( a s ( s
bc
2
? ? ? = 
bc
2 ?
Page # 42
6. Area of Triangle ( ?) :
? = 
2
1
ab sin
 
C = 
2
1
bc sin
 
A = 
2
1
ca sin
 
B = ss a sb sc ( )( )( ) ? ? ?
7. m
 
-
 
n Rule:
If BD : DC = m : n, then
(m
 
+
 
n) cot
 
? ? ?
? ?
m n
n B m C
cot cot
cot cot
? ?
8. Radius of Circumcirlce :
R =
C sin 2
c
B sin 2
b
A sin 2
a
? ?  = 
? 4
c b a
9. Radius of The Incircle :
(i) r = 
?
s
(ii) r = (s
 
?
 
a) tan
A
2
 = (s
 
?
 
b) tan
B
2
 = (s
 
?
 
c) tan
C
2
(iii) r = 4R sin
A
2
 sin
B
2
 sin
C
2
10. Radius of The Ex-
 
Circles :
(i) r
1
 =
?
s a ?
;
 r
2
 =
?
s b ?
;
 r
3
 =
 
?
s c ?
(ii)  r
1
 = s tan
A
2
;
  r
2
 = s tan
B
2
;
  r
3
 = s tan
C
2
(iii) r
1
 = 4 R sin
A
2
. cos
B
2
. cos
C
2
Page 3


Page # 41
SOLUTION OF TRIANGLE
1. Sine Rule:
C sin
c
B sin
b
A sin
a
? ? .
2. Cosine Formula:
(i) cos
 
A =
b c a
b c
2 2 2
2
? ?
   (ii)  cos
 
B = 
c a b
ca
222
2
? ?
(iii)  cos
 
C = 
a b c
a b
2 2 2
2
? ?
3. Projection  Formula:
(i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA
4. Napier’s Analogy - tangent rule:
(i) tan 
2
C B ?
 = 
c b
c b
?
?
 cot 
2
A
(ii)  tan
2
A C ?
 =
a c
a c
?
?
 cot
B
2
(iii)  tan
A B ?
2
 =
a b
a b
?
?
 cot
C
2
5. Trigonometric Functions of Half Angles:
(i) sin
A
2
 =
( ) ( ) s b s c
b c
? ?
 ; sin
B
2
 =
( ) ( ) s c s a
ca
? ?
 ;
sin
C
2
 = 
( ) ( ) s a s b
a b
? ?
(ii) cos
A
2
 =
s s a
b c
( ) ?
 ; cos
B
2
 =
s s b
ca
( ) ?
 ;  cos
C
2
 = 
s s c
a b
( ) ?
(iii) tan
A
2
 = 
( ) ( )
( )
s b s c
s s a
? ?
?
=
?
s s a ( ) ?
 where s =
a b c ? ?
2
 is semi
perimetre of triangle.
(iv) sin A = ) c s)( b s)( a s ( s
bc
2
? ? ? = 
bc
2 ?
Page # 42
6. Area of Triangle ( ?) :
? = 
2
1
ab sin
 
C = 
2
1
bc sin
 
A = 
2
1
ca sin
 
B = ss a sb sc ( )( )( ) ? ? ?
7. m
 
-
 
n Rule:
If BD : DC = m : n, then
(m
 
+
 
n) cot
 
? ? ?
? ?
m n
n B m C
cot cot
cot cot
? ?
8. Radius of Circumcirlce :
R =
C sin 2
c
B sin 2
b
A sin 2
a
? ?  = 
? 4
c b a
9. Radius of The Incircle :
(i) r = 
?
s
(ii) r = (s
 
?
 
a) tan
A
2
 = (s
 
?
 
b) tan
B
2
 = (s
 
?
 
c) tan
C
2
(iii) r = 4R sin
A
2
 sin
B
2
 sin
C
2
10. Radius of The Ex-
 
Circles :
(i) r
1
 =
?
s a ?
;
 r
2
 =
?
s b ?
;
 r
3
 =
 
?
s c ?
(ii)  r
1
 = s tan
A
2
;
  r
2
 = s tan
B
2
;
  r
3
 = s tan
C
2
(iii) r
1
 = 4 R sin
A
2
. cos
B
2
. cos
C
2
Page # 43
11. Length of Angle Bisectors, Medians & Altitudes :
(i)  Length of an angle bisector from the angle A = ?
a
 =
2
2
bc
b c
A
cos
?
 ;
(ii) Length of median from the angle A = m
a
 =
1
2
2 2
2 2 2
b c a ? ?
& (iii) Length of altitude from the angle A  = A
a
 =
a
2 ?
12. Orthocentre and Pedal Triangle:
The triangle KLM which is formed by joining the feet of the altitudes is
called the Pedal Triangle.
(i) Its angles are ? ? ? 2A, ? ? ? 2B and ? ? ? 2C.
(ii) Its sides are a cosA = R sin 2A,
 b cosB = R sin 2B   and
 c cosC = R sin 2C
(iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal.
Page 4


Page # 41
SOLUTION OF TRIANGLE
1. Sine Rule:
C sin
c
B sin
b
A sin
a
? ? .
2. Cosine Formula:
(i) cos
 
A =
b c a
b c
2 2 2
2
? ?
   (ii)  cos
 
B = 
c a b
ca
222
2
? ?
(iii)  cos
 
C = 
a b c
a b
2 2 2
2
? ?
3. Projection  Formula:
(i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA
4. Napier’s Analogy - tangent rule:
(i) tan 
2
C B ?
 = 
c b
c b
?
?
 cot 
2
A
(ii)  tan
2
A C ?
 =
a c
a c
?
?
 cot
B
2
(iii)  tan
A B ?
2
 =
a b
a b
?
?
 cot
C
2
5. Trigonometric Functions of Half Angles:
(i) sin
A
2
 =
( ) ( ) s b s c
b c
? ?
 ; sin
B
2
 =
( ) ( ) s c s a
ca
? ?
 ;
sin
C
2
 = 
( ) ( ) s a s b
a b
? ?
(ii) cos
A
2
 =
s s a
b c
( ) ?
 ; cos
B
2
 =
s s b
ca
( ) ?
 ;  cos
C
2
 = 
s s c
a b
( ) ?
(iii) tan
A
2
 = 
( ) ( )
( )
s b s c
s s a
? ?
?
=
?
s s a ( ) ?
 where s =
a b c ? ?
2
 is semi
perimetre of triangle.
(iv) sin A = ) c s)( b s)( a s ( s
bc
2
? ? ? = 
bc
2 ?
Page # 42
6. Area of Triangle ( ?) :
? = 
2
1
ab sin
 
C = 
2
1
bc sin
 
A = 
2
1
ca sin
 
B = ss a sb sc ( )( )( ) ? ? ?
7. m
 
-
 
n Rule:
If BD : DC = m : n, then
(m
 
+
 
n) cot
 
? ? ?
? ?
m n
n B m C
cot cot
cot cot
? ?
8. Radius of Circumcirlce :
R =
C sin 2
c
B sin 2
b
A sin 2
a
? ?  = 
? 4
c b a
9. Radius of The Incircle :
(i) r = 
?
s
(ii) r = (s
 
?
 
a) tan
A
2
 = (s
 
?
 
b) tan
B
2
 = (s
 
?
 
c) tan
C
2
(iii) r = 4R sin
A
2
 sin
B
2
 sin
C
2
10. Radius of The Ex-
 
Circles :
(i) r
1
 =
?
s a ?
;
 r
2
 =
?
s b ?
;
 r
3
 =
 
?
s c ?
(ii)  r
1
 = s tan
A
2
;
  r
2
 = s tan
B
2
;
  r
3
 = s tan
C
2
(iii) r
1
 = 4 R sin
A
2
. cos
B
2
. cos
C
2
Page # 43
11. Length of Angle Bisectors, Medians & Altitudes :
(i)  Length of an angle bisector from the angle A = ?
a
 =
2
2
bc
b c
A
cos
?
 ;
(ii) Length of median from the angle A = m
a
 =
1
2
2 2
2 2 2
b c a ? ?
& (iii) Length of altitude from the angle A  = A
a
 =
a
2 ?
12. Orthocentre and Pedal Triangle:
The triangle KLM which is formed by joining the feet of the altitudes is
called the Pedal Triangle.
(i) Its angles are ? ? ? 2A, ? ? ? 2B and ? ? ? 2C.
(ii) Its sides are a cosA = R sin 2A,
 b cosB = R sin 2B   and
 c cosC = R sin 2C
(iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal.
Page # 44
13.
The triangle formed by joining the three excentres ?
1
, ?
2
 and ?
3
 of ? ABC is
called the excentral or excentric triangle.
(i) ? ABC is the pedal triangle of the ? ?
1 
?
2 
?
3
.
(ii) Its angles are 
?
2 2
?
A
,
?
2 2
?
B
  &
?
2 2
?
C
.
(iii) Its sides are 4
 
R cos
A
2
, 4 R cos
B
2
 & 4 R cos
C
2
.
(iv) ? ? ?
1
 = 4
 
R sin
A
2
; ? ? ?
2
 = 4 R sin
B
2
; ? ? ?
3
 = 4 R sin
C
2
.
(v) Incentre ? of ? ABC is the orthocentre of the excentral  ? ?
1 
?
2 
?
3
.
14. Distance Between Special Points :
(i) Distance between circumcentre and orthocentre
OH
2
 = R
2
 (1 – 8 cosA cos B cos C)
(ii) Distance between circumcentre and incentre
O ?
2
 = R
2
 (1 – 8 sin
2
A
 sin
2
B
 sin
2
C
) = R
2
 – 2Rr
(iii) Distance between circumcentre and centroid
OG
2
 = R
2
 – 
9
1
(a
2
 + b
2
 + c
2
)