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   Final thoughts

4. Understanding the formulas--Page 3

In order to understand formulas very well, we should also know the relationships between some formulas. For example, we get the Sum and Difference formulas from the definitions of trigonometric functions. From Sum formulas, let alpha.gif (882 bytes) = beta.gif (942 bytes) = theta.gif (908 bytes), then we get the Double-Angle formulas. From Double-Angle formulas of cosine:

4_301.gif (2712 bytes)

we get Half-Angle formulas:

4_302.gif (3549 bytes)

In fact the relationship of alpha.gif (882 bytes) and alpha over 2 is double angle relationship. Sometimes, the following formulas:

4_303.gif (2828 bytes)

are very useful. Also we know:

4_304.gif (5117 bytes)

by adding these two formulas, we can get:

4_305.gif (3135 bytes) (1)          

Let 4_306.gif (1312 bytes), 4_307.gif (1274 bytes). If we add them together, we have 4_308.gif (1324 bytes), . From (1), we have: 4_310.gif (2866 bytes).

4_310.gif (2866 bytes) (2)          

After substituting x and y by alpha.gif (882 bytes) and beta.gif (942 bytes) respectively in (2), we have the Sum to Product formula:

4_311.gif (2866 bytes)

From (1) divide by 2 on both sides, we have the Product-to-Sum formula:

4_401.gif (3269 bytes)


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