Trigonometric transformation formulas





Trigonometric Transformation Formulas:

The set of formulas which are useful in transforming sums and difference of trigonometric functions into their products and vice versa.

These sets of formulas are derived directly from Trigonometric Addition and Subtraction formulas.

Here we will derive the transformation formulas using following four formulas which are Trigonometric addition and subtraction formulas:

\sin A \cos B + \cos A \sin B = \sin ( A + B ) \cdots{} i \\ \\ \sin A \cos B - \cos A \sin B = \sin ( A - B ) \cdots{} ii \\ \\ \cos A \cos B - \sin A \sin B = \cos ( A + B ) \cdots{} iii \\ \\ \cos A \cos B+ \sin A \sin B = \cos ( A - B ) \cdots{} iv

If we add and subtract the first two and last two equations in the equations above then we can get the following equations:

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

 

Now the the above four equation if we replace “A + B” by “C” and “A – B” by “D” or , A + B = C and A – B = D or ,  A = \frac{1}{2} (C + D) and B = \frac{1}{2} (C - D) then we can get the following equations:

 

\sin C + \sin D = 2 \sin \frac{1}{2}( C + D) \cos \frac{1}{2}( C - D) \\ \\ \sin C - \sin D = 2 \sin \frac{1}{2}( C - D) \cos \frac{1}{2}( C + D) \\ \\ \cos C + \cos D = 2 \cos \frac{1}{2}( C + D) \cos \frac{1}{2}( C - D) \\ \\ \cos D - \cos C = 2 \sin \frac{1}{2}( C + D) \sin \frac{1}{2}( C - D)

 

The eight formulas derived here are called the Trigonometric transformation formulas and they can be used to transform the sums or difference of trigonometric functions to their products and products of trigonometric functions to their sums or difference.



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