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         Page 2
   Final thoughts

5. Summarizing--Page 2

The answer of the third question is no in general cases. In the discussion of Understanding the formulas section we know many formulas can help us transform sum to product. For examples, in Double Angle formulas,

5_201.gif (3012 bytes)

the process from left to right is transforming sum to product.

In the sum angles formula:

5_202.gif (3047 bytes)

The process from right to left is transforming sum to product. By using this process we can transform 5_203.gif (1463 bytes) into a product as follows.

5_204.gif (6498 bytes)

In general, we can transform 5_205.gif (1614 bytes) to be product:

5_206.gif (8406 bytes)

where phi.gif (942 bytes) satisfies the system:

5_207.gif (3017 bytes)

or the equation 5_208.gif (1402 bytes).

From the examples above, we can see that our abilities in solving problems will be enhanced if we can summarize all formulas by the transformations of angles, functions, and operations.


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