COMPOUND ANGLES SUM AND DIFFERENCE

In this page compound angles sum and differences we are going to see combination of two formulas in compound angles.

We already know these two formulas

sin (A+B)  =  sin A cos B + cos A sin B -----(1)

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

by adding (1) + (2) we will get the new formula

sin(A+B) + sin (A-B)

  =  sin A cos B + cos A sin B + sin A cos B - cos A sin B

                          =  2sin A cos B

The new formula is sin(A+B) + sin (A-B)  =  2sin A cos B

by subtracting (1) - (2) we will get the new formula

Sin(A+B)-Sin (A-B)

  =  sin A cos B + cos A sin B - [sin A cos B - cos A sin B]

Sin(A+B)-Sin (A-B)

  =  sin A cos B + cos A sin B - sin A cos B + cos A sin B

                         =  cos A sin B + cos A sin B

                          = 2 cos A sin B

So the new formula is

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

cos (A+B)  =  cos A co

s B - sin A sin B -----(1)

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

by adding (1) + (2) we will get the new formula

cos(A+B) + cos (A-B)

  = cos A cos B - sin A sin B + cos A cos B + sin A sin B

  =  2 cos A cos B

So the new formula is

cos(A+B)+cos (A-B)  =  2 cos A cos B

by subtracting (1) - (2) we will get the new formula

cos(A+B)-cos (A-B)

  =  cos A cos B - sin A sin B-[cos A cos B + sin A sin B]

  =  cos A cos B-sin A sin B-cos A cos B-sin A sin B

                            = -2sin A sin B

So the new formula is

cos(A+B)-cos (A-B)  =  -2sin A sin B

The new derived formulas are

sin(A+B)+sin (A-B)  =  2 sin A cos B

sin(A+B)-sin (A-B)  =  2 cos A sin B

cos(A+B)+cos (A-B)  =  2 cos A cos B

cos(A+B)-cos (A-B)  =  -2sin A sin B

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