In any `triangle ABC, (sin A)/a = (sin B)/b = (sin C)/c ` i. e. the sines of angles are proportional to the lengths of the opposite sides.

Let `a/(sin A) = b/(sin B) = c/(sin C)= K` (say)

Then `a = K sin A, b = K sin B, c = K sin C` or `sin A = 1/K a, sin B =1/K b, sin C = 1/K c`

In any `triangle ABC`

(i) `a^2=b^2+c^2- 2bc cos A ` or `cos a= (b^2+c^2-a^2)/(2bc)`

(ii) `b^2=c^2+a^2- 2 ac cos B` or `cos B=(a^2+c^2-b^2)/(2ac)`

(iii) `c^2= a^2+b^2- 2ab cos C` or `cos C=(a^2+b^2-c^2)/(2ab)`

In any `Delta ABC`,

(i) `a= bcos C + c cos B`

(ii) `b = ccos A +a cos C`

(iii) `c= a cos B + bcos A`

Note Any side of a triangle is equal to the sum of the projections of other two sides on it.

In any `Delta ABC`, let `s =(a+ b +c) //2`, then

(i) `sin \ A/2 = sqrt(((s-b)(s-c))/(bc))`

(ii) `sin \ B/2 = sqrt(((s-a)(s-c))/(ac))`

(iii) `sin \ C/2 =sqrt(((s-a)(s-b))/(ab))`

(iv) `cos \ A/2 = sqrt(((s-b)(s-c))/(bc))`

(v) `cos \ B/2 = sqrt((s(s-b))/(ab))`

(vi) `cos \ C/2 = sqrt((s(s-c))/(ab) `

(vii) `tan \ A/2 = sqrt(((s-b) (s-c))/(s(s-a)))`

(viii) `tan \ B/2 = sqrt(((s-a)(s-c))/(s(s-b)))`

(ix) `tan \ C/2 = sqrt(((s-a)(s-b))/(s(s-c)))`

In any `triangle ABC`, by `sin 2 theta = 2sin theta * cos theta`

(i) `sin A = 2/(bc) sqrt(s(s-a)(s-b)(s-c))= (2 Delta)/(bc)`

(ii) `sin B = 2/(ac) sqrt(s(s-a)(s-b)(s-c))= (2 Delta)/(ac)`

(iii) `sin C = 2/(ab) sqrt(s(s-a) (s-b)(s-c))= (2 Delta)/(ab)`

where, `Delta=` area of scalene triangle, when `a ne b ne c`.

In a `Delta ABC`, if the sides of the triangle are `a, b, c` and corresponding angles are `A, B` and `C` respectively, then
area of triangle

(i) `Delta = 1/2 ab sin C , Delta = 1/2 bc sin A, Delta = 1/2 ca sin B`

(ii) `Delta =(c^2 sin A sin B)/(2 sin B), Delta =(a^2 sin B sin C)/(2 sin A) , Delta = (b^2 sin C sin A)/(2 sin B)`

(iii) `Delta = sqrt(s(s-a) (s-b) (s-c))`

In any `triangle ABC`,

(i) `tan ((B-C)/2) =((b-c)/(b+c)) cot \ A/2`

(ii) `tan ((C-A)/2) =((c-a)/(c+a)) cot \ B/2`

(iii) `tan ((A-B)/2) = ((a-b)/(a+b)) cot \ C/2`

A circle passing through the vertices of a `Delta ABC` is called circumcircle and its radius is called circumradius (R )
and its centre is known as circumcentre.

Circumradius of triangle is given by

`R= a/( 2sin A) = b/(2 sin B) = c/( 2sin C) =(abc)/(4 Delta)`

The circle which can be inscribed within the triangle so as to touch each of its sides is called its inscribed circle
or incircle and its centre is called incentre and its radius is called inradius (`r`).

In any `Delta ABC`,

(i) `r= Delta/s`

(ii) `r=(s-a) tan \ A/2 =(s-b) tan \ B/2 =(s-c) tan \ C/2`

(iii) `r= 4 R sin \ A/2 sin \ B/2 sin \ C/2`

(iv) `r= (a sin B//2 sin C//2)/(cos A//2)`

`= (b sin A//2 sin C//2)/(cos B//2)`

`= (c sin B//2 sin A//2)/(cos C//2)`

The circle which touches the sides `BC` and two sides `AB` and `A C` produced of a `triangle ABC`, is called the escribed
circle opposite to the `angle A`. Its radius is denoted by `r_1`.

Similarly, `r_2` and `r_3` denote the radii of the escribed circles opposite to the angles `B` and `C`, respectively

The centres of the escribed circles are call excentres.

In any `triangle ABC`,

(i) `r_1 =(Delta)/(s-a) = s tan \ A/2 = 4R sin \ A/2 cos \ B/2 cos \ C/2`

(ii) `r_2 = Delta/(s-b) = s tan \ B/2 = 4R sin \ B/2 cos \ C/2 cos \ A/2`

(iii) `r_3 = Delta/(s-c)= s tan \ C/2 = 4 R sin \ C/2 cos \ A/2 cos \ B/2`

(iv) `r_1 +r_2 +r_3 = 4R + r`

(v) `r_1r_2 +r_2 r_3 +r_3 r_1 = s^2 =(r_1r_2r_3)/r`

(vi) `r_1 =(a cos B//2 cos C//2)/(cos A//2)`

(vii) `r_2 =(b cos C//2 cos A//2)/(cos B//2)`

(viii) `r_3 =(c cos A//2 cos B//2)/(cos C//2)`

(ix) `1/r_1 + 1/r_2 + 1/r_3 = s/Delta = 1/r`

and `r_1r_2r_3 = r^2 (cot \ A/2 cot \ B/2 cot \ C/2)^2`