Half angle formulae or Semi-sum formulae

Let s be the half perimeter of the ABC i.e. 2 s = a +b +c, then

(i)   (ii)

(iii) tan A/2 =

Area of a Triangle

In a triangle ABC, the area is given by
      =(1/2)bc sin A = (1/2)ca sin B = (1/2)ab sin C

Heron formula

Illustrative Examples

Example

In a ABC, a = 3, b = 5, c = 6. Calculate

(i) sin A     (ii) cos A    (iii) tan A    (iv) sin A/2   (v) cos A/2
(vi) tan A/2                (vii) area of triangle i.e.

Solution

Here 2s = a +b +c = 3 +5 +6 = 14,
So s = 7, s -a = 4, s - b = 2, s -c = 1

1. sin A = 2/bc= [2 s (s -a)(s -b)(s -c)]/bc = (2 7.4.2.1)/(5.6) = (2 14)/15
2. cos A =[ b² +c² -a²] / (2bc) = [5² +6² -3²] / (2.5.6) = [25 +36 -9]/60 = 52/60 = 13/15
3. tan A = sin A/cos A = (214)/13
4.   = 1/15
5. = 1/15
6.  
7. = s (s -a)(s -b)(s -c) = 7.4.2.1 = 2 14

Exercise

In a ABC, prove that (1 -5) :

1. = [a² sin B sin C]/ 2 sinA = [b² sin C sin A]/2 sinB = [c² sin A sin B]/2 sinC
2. = s (s -a) tan A/2 = s(s -b) tan B/2 = s (s -c) tan C/2
3. (i) cot A/2 +cot B/2 +cot C/2 = s²/
   (ii) tan A/2 tan B/2 tan C/2 = /s²
4. tan A/2 tan B/2 = (a +b -c)/(a +b +c) = (s -c)/s
5. a² sin 2 B +b² sin 2 A = 4
6. If a = 18, b = 24, c = 30, calculate the following :
   (i)             (ii) sin A        (iii) cos A         (iv) tan A
   (v) sin A/2  (vi) cos A/2    (vii) tan A/2

Answers