Important Formulas: Solutions of Triangles

# Important Solutions of Triangles Formulas for JEE and NEET

``` 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
? ?
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

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
? ?
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

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
? ?
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

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
)
```

## Mathematics (Maths) for JEE Main & Advanced

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## Mathematics (Maths) for JEE Main & Advanced

184 videos|577 docs|187 tests

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