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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