Mathematics SUM OR DIFFERENCE OF THE ANGLE

SUM OR DIFFERENCE OF THE ANGLE

The algebraic sums of two or more angles are generally called compound angles and the angles are known as the constituent angles.
For example : If `A, B, C` are three angles then `A - B, A + B + C, A - B + C` etc. are compound angles.

`(a) \ \ \ \ \ \sin (A + B) =sin A cos B + cos A sin B`

`(b) \ \ \ \ \ \sin(A - B) = sinA cosB - cosA sinB`

`(c) \ \ \ \ \ \cos (A + B) = cos A cos B - sin A sin B`

`(d) \ \ \ \ \ \cos (A - B) = cos A cos B + sin A sin B`

`(e) \ \ \ \ \ \tan (A + B) = (tanA+ tanB) /(1-tanA tanB)`

`(f) \ \ \ \ \ \ tan (A - B) =(tanA- tanB) /(1+tanA tanB)`

`(g) \ \ \ \ \ \cot (A + B) = (cotA cot B-1)/(cotB + cotA)`

`(h) \ \ \ \ \ \cot (A - B) =(cotA cot B+1)/(cotB - cotA)`

Some More Results :

`(a) \ \ \ \ \ \sin (A + B).sin (A - B) = sin^2 A - sin^2 B = cos^2 B - cos^2 A`

`(b)\ \ \ \ \ \ cos (A + B).cos (A - B) = cos^2 A - sin^2 B= cos^2 B - sin^2 A`

`(c) \ \ \ \ \ \sin (A + B + C) = sin A cos B cos C + cos A sin B sin C + cos A cos B sin C - sin A sin B sin C`

`(d) \ \ \ \ \ \cos (A + B + C) = cos A cos B cos C - cos A. sin B sin C - sin A cos B sin C - sin A sinB cosC`

`(e) \ \ \ \ \ \tan (A + B + C)=(tanA+ tanB+ tan C-tanA tanB tanC)/(1-tanA tanB- tanB tanC -tanC tanA)`

FORMULA TO TRANSFORM THE PRODUCT INTO SUM OR DIFFERENCE

We know that,
`sin A cos B + cos A sin B = sin (A + B) .......(i)`

`sin A cos B - cos A sin B = sin (A - B) ......(ii)`

`cos A cos B - sin A sin B = cos (A + B) .....(iii)`

`cos A cos B + sin A sin B = cos (A - B) .....(iv)`

Adding `(i)` and `(ii),`

`2 sin A cos B = sin (A + B) + sin (A - B)`

Subtracting `(ii)` from `(i),`

`2 cos A sin B = sin (A + B) - sin (A - B)`

Adding `(iii)` and `(iv),`

`2 cosA cos B = cos (A + B) + cos (A - B)`

Subtraction `(iii)` from `(iv).`

`2 sin A sin B = cos (A - B) - cos (A + B)`

Formula :
`(a) \ \ \ \ \ \2 sin A cos B = sin (A + B) + sin (A - B)`

`(b)\ \ \ \ \ \ 2 cos A sin B = sin (A + B) - sin (A - B)`

`(c) \ \ \ \ \ \2 cos A cos B = cos (A + B) + cos (A - B)`

`(d) \ \ \ \ \ \2 sin A sin B = cos (A - B) - cos (A + B)`

FORMULA TO TRANSFORM THE SUM OR DIFFERENCE INTO PRODUCT

We know that,
`sin (A + B) + sin(A - B) = 2 sin A cos B ......(i)`

Let `A+ B = C` and `A - B = D`

then `A =((C +D)/2)` and `B =((C -D)/2)`

Substituting in `(i),`

`(a)\ \ \ \ \ \ sin C+ sin D = 2 sin((C +D)/2).cos((C -D)/2)`

similarly other formula,

`(b) \ \ \ \ \ \sin C - sin D = 2 cos ((C +D)/2).sin ((C -D)/2) `

`(c)\ \ \ \ \ \ cos C + cos D= 2 cos ((C +D)/2) .cos ((C -D)/2) `

`(d)\ \ \ \ \ \ cos C - cos D = 2 sin ((C +D)/2) .sin ((C -D)/2) `